Abstract

Multimode interference reflectors (MIRs) were recently introduced as a new type of photonic integrated devices for on-chip, broadband light reflection. In the original proposal, different MIRs were demonstrated based on total internal reflection mirrors made of two deep-etched facets. Although simpler to fabricate, this approach imposes certain limits on the shape of the field pattern at the reflecting facets, which in turn restricts the types of MIRs that can be implemented. In this work, we propose and experimentally demonstrate the use of aluminium-based mirrors for the design of 2-port MIRs with variable reflectivity. These mirrors do not impose any restrictions on the incident field, and thus give more flexibility at the design stage. Devices with different reflectivities in the range between 0 and 0.5 were fabricated in a 3 um thick SOI platform, and characterization of multiple dies was performed to extract statistical data about their performance. Our measurements show that, on average, losses both in the aluminium mirror and in the access waveguides reduce the reflectivities to about 79% of their target value. Moreover, standard deviations lower than ±5% are obtained over a 20 nm wavelength range (1540–1560 nm). We also provide a theoretical model of the aluminium mirror based on the effective index method and Fresnel equations in multilayer thin films, which shows good agreement with FDTD simulations.

© 2015 Optical Society of America

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    [Crossref]

2015 (2)

2014 (3)

B. Gargallo, P. Muñoz, R. Baños, A. L. Giesecke, J. Bolten, T. Wahlbrink, and H. Kleinjans, “Reflective arrayed waveguide gratings based on Sagnac loop reflectors with custom spectral response,” Opt. Express 22, 14348–14362 (2014).
[Crossref] [PubMed]

E. Kleijn, D. Melati, A. Melloni, T. de Vries, M. Smit, and X. Leijtens, “Multimode Interference Couplers With Reduced Parasitic Reflections,” IEEE Photon. Technol. Lett. 26, 408–410 (2014).
[Crossref]

J. D. Doménech, J. S. Fandiño, B. Gargallo, and P. Muñoz, “Arbitrary Coupling Ratio Multimode Interference Couplers in Silicon-on-Insulator,” J. Lightw. Technol. 32, 2536–2543 (2014).
[Crossref]

2013 (3)

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

A. Maese-Novo, R. Halir, S. Romero-García, D. Pérez-Galacho, L. Zavargo-Peche, A. O.-M. nux, I. Molina-Fernández, J. G. Wangüemert-Pérez, and P. Cheben, “Wavelength independent multimode interference coupler,” Opt. Express 21, 7033–7040 (2013).
[Crossref] [PubMed]

E. Kleijn, M. Smit, and X. Leijtens, “Multimode Interference Reflectors: A New Class of Components for Photonic Integrated Circuits,” J. Lightw. Technol. 31, 3055–3063 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[Crossref]

2009 (1)

T. Fukuda, K. Okamoto, Y. Hinokuma, and K. Hamamoto, “Phase-Locked Array Laser Diodes (LDs) by Using 1xN Active Multimode-Interferometer (MMI),” IEEE Photon. Technol. Lett. 21, 176–178 (2009).
[Crossref]

2006 (4)

H. Chen and A. Poon, “Low-Loss Multimode-Interference-Based Crossings for Silicon Wire Waveguides,” IEEE Photon. Technol. Lett. 18, 2260–2262 (2006).
[Crossref]

F. Wang, J. Yang, L. Chen, X. Jiang, and M. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. 18, 421–423 (2006).
[Crossref]

K. Solehmainen, T. Aalto, J. Dekker, M. Kapulainen, M. Harjanne, and P. Heimala, “Development of multi-step processing in Silicon-on-Insulator for optical waveguide applications,” J. Opt. Pure Appl. Opt. 8, S455–S460 (2006).
[Crossref]

M. Seimetz and C.-M. Weinert, “Options, Feasibility, and Availability of 2×4 90° Hybrids for Coherent Optical Systems,” J. Lightw. Technol. 24, 1317 (2006).
[Crossref]

2001 (1)

J. Azana and M. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Topics Quantum Electron. 7, 728–744 (2001).
[Crossref]

2000 (1)

A. E. Kaplan, I. Marzoli, W. E. Lamb, and W. P. Schleich, “Multimode interference: Highly regular pattern formation in quantum wave-packet evolution,” Phys. Rev. A 61, 032101 (2000).
[Crossref]

1996 (2)

P. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightw. Technol. 14, 2286–2293 (1996).
[Crossref]

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” Journal of Modern Optics 43, 2139–2164 (1996).
[Crossref]

1995 (2)

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

M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34, 6898–6910 (1995).
[Crossref] [PubMed]

1994 (2)

E. Pennings, R. van Roijen, M. van Stralen, P. de Waard, R. Koumans, and B. Verbeck, “Reflection properties of multimode interference devices,” IEEE Photon. Technol. Lett. 6, 715–718 (1994).
[Crossref]

P. Besse, M. Bachmann, H. Melchior, L. Soldano, and M. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightw. Technol. 12, 1004–1009 (1994).
[Crossref]

Aalto, T.

M. Cherchi, S. Ylinen, M. Harjanne, M. Kapulainen, and T. Aalto, “MMI resonators based on metal mirrors and MMI mirrors: an experimental comparison,” Opt. Express 23, 5982–5993 (2015).
[Crossref] [PubMed]

K. Solehmainen, T. Aalto, J. Dekker, M. Kapulainen, M. Harjanne, and P. Heimala, “Development of multi-step processing in Silicon-on-Insulator for optical waveguide applications,” J. Opt. Pure Appl. Opt. 8, S455–S460 (2006).
[Crossref]

Alonso-Ramos, C.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Azana, J.

J. Azana and M. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Topics Quantum Electron. 7, 728–744 (2001).
[Crossref]

Bach, H.-G.

R. Kunkel, H.-G. Bach, D. Hoffmann, C. Weinert, I. Molina-Fernandez, and R. Halir, “First monolithic InP-based 90°-hybrid OEIC comprising balanced detectors for 100GE coherent frontends,” in “IEEE International Conference on Indium Phosphide Related Materials (IPRM),” (2009), pp. 167–170.

Bachmann, M.

P. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightw. Technol. 14, 2286–2293 (1996).
[Crossref]

M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34, 6898–6910 (1995).
[Crossref] [PubMed]

P. Besse, M. Bachmann, H. Melchior, L. Soldano, and M. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightw. Technol. 12, 1004–1009 (1994).
[Crossref]

Baños, R.

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[Crossref]

Berry, M. V.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” Journal of Modern Optics 43, 2139–2164 (1996).
[Crossref]

Besse, P.

P. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightw. Technol. 14, 2286–2293 (1996).
[Crossref]

P. Besse, M. Bachmann, H. Melchior, L. Soldano, and M. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightw. Technol. 12, 1004–1009 (1994).
[Crossref]

Besse, P. A.

Bolten, J.

Carpintero, G.

Cheben, P.

A. Maese-Novo, R. Halir, S. Romero-García, D. Pérez-Galacho, L. Zavargo-Peche, A. O.-M. nux, I. Molina-Fernández, J. G. Wangüemert-Pérez, and P. Cheben, “Wavelength independent multimode interference coupler,” Opt. Express 21, 7033–7040 (2013).
[Crossref] [PubMed]

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Chen, H.

H. Chen and A. Poon, “Low-Loss Multimode-Interference-Based Crossings for Silicon Wire Waveguides,” IEEE Photon. Technol. Lett. 18, 2260–2262 (2006).
[Crossref]

Chen, L.

F. Wang, J. Yang, L. Chen, X. Jiang, and M. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. 18, 421–423 (2006).
[Crossref]

Chen, R. T.

Cherchi, M.

Covey, J.

de Vries, T.

E. Kleijn, D. Melati, A. Melloni, T. de Vries, M. Smit, and X. Leijtens, “Multimode Interference Couplers With Reduced Parasitic Reflections,” IEEE Photon. Technol. Lett. 26, 408–410 (2014).
[Crossref]

L. Xu, X. Leijtens, B. Docter, T. de Vries, E. Smalbrugge, F. Karouta, and M. Smit, “MMI-reflector: A novel on-chip reflector for photonic integrated circuits,” in “European Conference on Optical Communications (ECOC),” (2009), pp. 1–2.

de Waard, P.

E. Pennings, R. van Roijen, M. van Stralen, P. de Waard, R. Koumans, and B. Verbeck, “Reflection properties of multimode interference devices,” IEEE Photon. Technol. Lett. 6, 715–718 (1994).
[Crossref]

Dekker, J.

K. Solehmainen, T. Aalto, J. Dekker, M. Kapulainen, M. Harjanne, and P. Heimala, “Development of multi-step processing in Silicon-on-Insulator for optical waveguide applications,” J. Opt. Pure Appl. Opt. 8, S455–S460 (2006).
[Crossref]

Docter, B.

L. Xu, X. Leijtens, B. Docter, T. de Vries, E. Smalbrugge, F. Karouta, and M. Smit, “MMI-reflector: A novel on-chip reflector for photonic integrated circuits,” in “European Conference on Optical Communications (ECOC),” (2009), pp. 1–2.

Doménech, J. D.

J. D. Doménech, J. S. Fandiño, B. Gargallo, and P. Muñoz, “Arbitrary Coupling Ratio Multimode Interference Couplers in Silicon-on-Insulator,” J. Lightw. Technol. 32, 2536–2543 (2014).
[Crossref]

Fandiño, J. S.

J. D. Doménech, J. S. Fandiño, B. Gargallo, and P. Muñoz, “Arbitrary Coupling Ratio Multimode Interference Couplers in Silicon-on-Insulator,” J. Lightw. Technol. 32, 2536–2543 (2014).
[Crossref]

Finlayson, E. D.

Fukuda, T.

T. Fukuda, K. Okamoto, Y. Hinokuma, and K. Hamamoto, “Phase-Locked Array Laser Diodes (LDs) by Using 1xN Active Multimode-Interferometer (MMI),” IEEE Photon. Technol. Lett. 21, 176–178 (2009).
[Crossref]

Gardes, F. Y.

Gargallo, B.

B. Gargallo, P. Muñoz, R. Baños, A. L. Giesecke, J. Bolten, T. Wahlbrink, and H. Kleinjans, “Reflective arrayed waveguide gratings based on Sagnac loop reflectors with custom spectral response,” Opt. Express 22, 14348–14362 (2014).
[Crossref] [PubMed]

J. D. Doménech, J. S. Fandiño, B. Gargallo, and P. Muñoz, “Arbitrary Coupling Ratio Multimode Interference Couplers in Silicon-on-Insulator,” J. Lightw. Technol. 32, 2536–2543 (2014).
[Crossref]

Giesecke, A. L.

Gini, E.

P. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightw. Technol. 14, 2286–2293 (1996).
[Crossref]

Gordón, C.

Guzmán, R.

Halir, R.

A. Maese-Novo, R. Halir, S. Romero-García, D. Pérez-Galacho, L. Zavargo-Peche, A. O.-M. nux, I. Molina-Fernández, J. G. Wangüemert-Pérez, and P. Cheben, “Wavelength independent multimode interference coupler,” Opt. Express 21, 7033–7040 (2013).
[Crossref] [PubMed]

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

R. Kunkel, H.-G. Bach, D. Hoffmann, C. Weinert, I. Molina-Fernandez, and R. Halir, “First monolithic InP-based 90°-hybrid OEIC comprising balanced detectors for 100GE coherent frontends,” in “IEEE International Conference on Indium Phosphide Related Materials (IPRM),” (2009), pp. 167–170.

Hamamoto, K.

T. Fukuda, K. Okamoto, Y. Hinokuma, and K. Hamamoto, “Phase-Locked Array Laser Diodes (LDs) by Using 1xN Active Multimode-Interferometer (MMI),” IEEE Photon. Technol. Lett. 21, 176–178 (2009).
[Crossref]

Harjanne, M.

M. Cherchi, S. Ylinen, M. Harjanne, M. Kapulainen, and T. Aalto, “MMI resonators based on metal mirrors and MMI mirrors: an experimental comparison,” Opt. Express 23, 5982–5993 (2015).
[Crossref] [PubMed]

K. Solehmainen, T. Aalto, J. Dekker, M. Kapulainen, M. Harjanne, and P. Heimala, “Development of multi-step processing in Silicon-on-Insulator for optical waveguide applications,” J. Opt. Pure Appl. Opt. 8, S455–S460 (2006).
[Crossref]

Hashizume, Y.

K. Takiguchi, T. Kitoh, M. Oguma, Y. Hashizume, and H. Takahashi, “Integrated-optic OFDM Demultiplexer using Multi-mode Interference Coupler-based Optical DFT Circuit,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2012), p. OM3J.6.

Heimala, P.

K. Solehmainen, T. Aalto, J. Dekker, M. Kapulainen, M. Harjanne, and P. Heimala, “Development of multi-step processing in Silicon-on-Insulator for optical waveguide applications,” J. Opt. Pure Appl. Opt. 8, S455–S460 (2006).
[Crossref]

Hinokuma, Y.

T. Fukuda, K. Okamoto, Y. Hinokuma, and K. Hamamoto, “Phase-Locked Array Laser Diodes (LDs) by Using 1xN Active Multimode-Interferometer (MMI),” IEEE Photon. Technol. Lett. 21, 176–178 (2009).
[Crossref]

Hoffmann, D.

R. Kunkel, H.-G. Bach, D. Hoffmann, C. Weinert, I. Molina-Fernandez, and R. Halir, “First monolithic InP-based 90°-hybrid OEIC comprising balanced detectors for 100GE coherent frontends,” in “IEEE International Conference on Indium Phosphide Related Materials (IPRM),” (2009), pp. 167–170.

Hosseini, A.

Hu, Y.

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[Crossref]

Janz, S.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Jenkins, R. M.

Jiang, X.

F. Wang, J. Yang, L. Chen, X. Jiang, and M. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. 18, 421–423 (2006).
[Crossref]

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[Crossref]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[Crossref]

Kaplan, A. E.

A. E. Kaplan, I. Marzoli, W. E. Lamb, and W. P. Schleich, “Multimode interference: Highly regular pattern formation in quantum wave-packet evolution,” Phys. Rev. A 61, 032101 (2000).
[Crossref]

Kapulainen, M.

M. Cherchi, S. Ylinen, M. Harjanne, M. Kapulainen, and T. Aalto, “MMI resonators based on metal mirrors and MMI mirrors: an experimental comparison,” Opt. Express 23, 5982–5993 (2015).
[Crossref] [PubMed]

K. Solehmainen, T. Aalto, J. Dekker, M. Kapulainen, M. Harjanne, and P. Heimala, “Development of multi-step processing in Silicon-on-Insulator for optical waveguide applications,” J. Opt. Pure Appl. Opt. 8, S455–S460 (2006).
[Crossref]

Karouta, F.

L. Xu, X. Leijtens, B. Docter, T. de Vries, E. Smalbrugge, F. Karouta, and M. Smit, “MMI-reflector: A novel on-chip reflector for photonic integrated circuits,” in “European Conference on Optical Communications (ECOC),” (2009), pp. 1–2.

Kitoh, T.

K. Takiguchi, T. Kitoh, M. Oguma, Y. Hashizume, and H. Takahashi, “Integrated-optic OFDM Demultiplexer using Multi-mode Interference Coupler-based Optical DFT Circuit,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2012), p. OM3J.6.

Kleijn, E.

E. Kleijn, D. Melati, A. Melloni, T. de Vries, M. Smit, and X. Leijtens, “Multimode Interference Couplers With Reduced Parasitic Reflections,” IEEE Photon. Technol. Lett. 26, 408–410 (2014).
[Crossref]

E. Kleijn, M. Smit, and X. Leijtens, “Multimode Interference Reflectors: A New Class of Components for Photonic Integrated Circuits,” J. Lightw. Technol. 31, 3055–3063 (2013).
[Crossref]

Klein, S.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” Journal of Modern Optics 43, 2139–2164 (1996).
[Crossref]

Kleinjans, H.

Koumans, R.

E. Pennings, R. van Roijen, M. van Stralen, P. de Waard, R. Koumans, and B. Verbeck, “Reflection properties of multimode interference devices,” IEEE Photon. Technol. Lett. 6, 715–718 (1994).
[Crossref]

Kunkel, R.

R. Kunkel, H.-G. Bach, D. Hoffmann, C. Weinert, I. Molina-Fernandez, and R. Halir, “First monolithic InP-based 90°-hybrid OEIC comprising balanced detectors for 100GE coherent frontends,” in “IEEE International Conference on Indium Phosphide Related Materials (IPRM),” (2009), pp. 167–170.

Kwong, D.

Lamb, W. E.

A. E. Kaplan, I. Marzoli, W. E. Lamb, and W. P. Schleich, “Multimode interference: Highly regular pattern formation in quantum wave-packet evolution,” Phys. Rev. A 61, 032101 (2000).
[Crossref]

Lapointe, J.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Leijtens, X.

C. Gordón, R. Guzmán, X. Leijtens, and G. Carpintero, “On-chip mode-locked laser diode structure using multimode interference reflectors,” Photon. Res. 3, 15–18 (2015).
[Crossref]

E. Kleijn, D. Melati, A. Melloni, T. de Vries, M. Smit, and X. Leijtens, “Multimode Interference Couplers With Reduced Parasitic Reflections,” IEEE Photon. Technol. Lett. 26, 408–410 (2014).
[Crossref]

E. Kleijn, M. Smit, and X. Leijtens, “Multimode Interference Reflectors: A New Class of Components for Photonic Integrated Circuits,” J. Lightw. Technol. 31, 3055–3063 (2013).
[Crossref]

L. Xu, X. Leijtens, B. Docter, T. de Vries, E. Smalbrugge, F. Karouta, and M. Smit, “MMI-reflector: A novel on-chip reflector for photonic integrated circuits,” in “European Conference on Optical Communications (ECOC),” (2009), pp. 1–2.

Maese-Novo, A.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

A. Maese-Novo, R. Halir, S. Romero-García, D. Pérez-Galacho, L. Zavargo-Peche, A. O.-M. nux, I. Molina-Fernández, J. G. Wangüemert-Pérez, and P. Cheben, “Wavelength independent multimode interference coupler,” Opt. Express 21, 7033–7040 (2013).
[Crossref] [PubMed]

Marzoli, I.

A. E. Kaplan, I. Marzoli, W. E. Lamb, and W. P. Schleich, “Multimode interference: Highly regular pattern formation in quantum wave-packet evolution,” Phys. Rev. A 61, 032101 (2000).
[Crossref]

Mashanovich, G. Z.

Melati, D.

E. Kleijn, D. Melati, A. Melloni, T. de Vries, M. Smit, and X. Leijtens, “Multimode Interference Couplers With Reduced Parasitic Reflections,” IEEE Photon. Technol. Lett. 26, 408–410 (2014).
[Crossref]

Melchior, H.

P. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightw. Technol. 14, 2286–2293 (1996).
[Crossref]

M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34, 6898–6910 (1995).
[Crossref] [PubMed]

P. Besse, M. Bachmann, H. Melchior, L. Soldano, and M. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightw. Technol. 12, 1004–1009 (1994).
[Crossref]

Melloni, A.

E. Kleijn, D. Melati, A. Melloni, T. de Vries, M. Smit, and X. Leijtens, “Multimode Interference Couplers With Reduced Parasitic Reflections,” IEEE Photon. Technol. Lett. 26, 408–410 (2014).
[Crossref]

Molina-Fernandez, I.

R. Kunkel, H.-G. Bach, D. Hoffmann, C. Weinert, I. Molina-Fernandez, and R. Halir, “First monolithic InP-based 90°-hybrid OEIC comprising balanced detectors for 100GE coherent frontends,” in “IEEE International Conference on Indium Phosphide Related Materials (IPRM),” (2009), pp. 167–170.

Molina-Fernández, I.

Molina-Fernndez, I.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Muñoz, P.

J. D. Doménech, J. S. Fandiño, B. Gargallo, and P. Muñoz, “Arbitrary Coupling Ratio Multimode Interference Couplers in Silicon-on-Insulator,” J. Lightw. Technol. 32, 2536–2543 (2014).
[Crossref]

B. Gargallo, P. Muñoz, R. Baños, A. L. Giesecke, J. Bolten, T. Wahlbrink, and H. Kleinjans, “Reflective arrayed waveguide gratings based on Sagnac loop reflectors with custom spectral response,” Opt. Express 22, 14348–14362 (2014).
[Crossref] [PubMed]

Muriel, M.

J. Azana and M. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Topics Quantum Electron. 7, 728–744 (2001).
[Crossref]

nux, A. O.-M.

Oguma, M.

K. Takiguchi, T. Kitoh, M. Oguma, Y. Hashizume, and H. Takahashi, “Integrated-optic OFDM Demultiplexer using Multi-mode Interference Coupler-based Optical DFT Circuit,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2012), p. OM3J.6.

Okamoto, K.

T. Fukuda, K. Okamoto, Y. Hinokuma, and K. Hamamoto, “Phase-Locked Array Laser Diodes (LDs) by Using 1xN Active Multimode-Interferometer (MMI),” IEEE Photon. Technol. Lett. 21, 176–178 (2009).
[Crossref]

Orfanidis, S. J.

S. J. Orfanidis, Electromagnetic Waves and Antennas (Rutgers University, 2008), chap. 8, pp. 305–307.

Ortega-Moux, A.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[Crossref]

Pennings, E.

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

E. Pennings, R. van Roijen, M. van Stralen, P. de Waard, R. Koumans, and B. Verbeck, “Reflection properties of multimode interference devices,” IEEE Photon. Technol. Lett. 6, 715–718 (1994).
[Crossref]

Pérez-Galacho, D.

Poon, A.

H. Chen and A. Poon, “Low-Loss Multimode-Interference-Based Crossings for Silicon Wire Waveguides,” IEEE Photon. Technol. Lett. 18, 2260–2262 (2006).
[Crossref]

Prez-Galacho, D.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Rahimi, S.

Reed, G. T.

Romero-García, S.

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd Edition (Wiley, 2007), chap. 8, pp. 299–308.

Schleich, W. P.

A. E. Kaplan, I. Marzoli, W. E. Lamb, and W. P. Schleich, “Multimode interference: Highly regular pattern formation in quantum wave-packet evolution,” Phys. Rev. A 61, 032101 (2000).
[Crossref]

Schmid, J. H.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Seimetz, M.

M. Seimetz and C.-M. Weinert, “Options, Feasibility, and Availability of 2×4 90° Hybrids for Coherent Optical Systems,” J. Lightw. Technol. 24, 1317 (2006).
[Crossref]

Smalbrugge, E.

L. Xu, X. Leijtens, B. Docter, T. de Vries, E. Smalbrugge, F. Karouta, and M. Smit, “MMI-reflector: A novel on-chip reflector for photonic integrated circuits,” in “European Conference on Optical Communications (ECOC),” (2009), pp. 1–2.

Smit, M.

E. Kleijn, D. Melati, A. Melloni, T. de Vries, M. Smit, and X. Leijtens, “Multimode Interference Couplers With Reduced Parasitic Reflections,” IEEE Photon. Technol. Lett. 26, 408–410 (2014).
[Crossref]

E. Kleijn, M. Smit, and X. Leijtens, “Multimode Interference Reflectors: A New Class of Components for Photonic Integrated Circuits,” J. Lightw. Technol. 31, 3055–3063 (2013).
[Crossref]

P. Besse, M. Bachmann, H. Melchior, L. Soldano, and M. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightw. Technol. 12, 1004–1009 (1994).
[Crossref]

L. Xu, X. Leijtens, B. Docter, T. de Vries, E. Smalbrugge, F. Karouta, and M. Smit, “MMI-reflector: A novel on-chip reflector for photonic integrated circuits,” in “European Conference on Optical Communications (ECOC),” (2009), pp. 1–2.

Soldano, L.

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

P. Besse, M. Bachmann, H. Melchior, L. Soldano, and M. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightw. Technol. 12, 1004–1009 (1994).
[Crossref]

Solehmainen, K.

K. Solehmainen, T. Aalto, J. Dekker, M. Kapulainen, M. Harjanne, and P. Heimala, “Development of multi-step processing in Silicon-on-Insulator for optical waveguide applications,” J. Opt. Pure Appl. Opt. 8, S455–S460 (2006).
[Crossref]

Takahashi, H.

K. Takiguchi, T. Kitoh, M. Oguma, Y. Hashizume, and H. Takahashi, “Integrated-optic OFDM Demultiplexer using Multi-mode Interference Coupler-based Optical DFT Circuit,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2012), p. OM3J.6.

Takiguchi, K.

K. Takiguchi, T. Kitoh, M. Oguma, Y. Hashizume, and H. Takahashi, “Integrated-optic OFDM Demultiplexer using Multi-mode Interference Coupler-based Optical DFT Circuit,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2012), p. OM3J.6.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd Edition (Wiley, 2007), chap. 8, pp. 299–308.

van Roijen, R.

E. Pennings, R. van Roijen, M. van Stralen, P. de Waard, R. Koumans, and B. Verbeck, “Reflection properties of multimode interference devices,” IEEE Photon. Technol. Lett. 6, 715–718 (1994).
[Crossref]

van Stralen, M.

E. Pennings, R. van Roijen, M. van Stralen, P. de Waard, R. Koumans, and B. Verbeck, “Reflection properties of multimode interference devices,” IEEE Photon. Technol. Lett. 6, 715–718 (1994).
[Crossref]

Verbeck, B.

E. Pennings, R. van Roijen, M. van Stralen, P. de Waard, R. Koumans, and B. Verbeck, “Reflection properties of multimode interference devices,” IEEE Photon. Technol. Lett. 6, 715–718 (1994).
[Crossref]

Wahlbrink, T.

Wang, F.

F. Wang, J. Yang, L. Chen, X. Jiang, and M. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. 18, 421–423 (2006).
[Crossref]

Wang, M.

F. Wang, J. Yang, L. Chen, X. Jiang, and M. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. 18, 421–423 (2006).
[Crossref]

Wangemert-Prez, J. G.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Wangüemert-Pérez, J. G.

Weinert, C.

R. Kunkel, H.-G. Bach, D. Hoffmann, C. Weinert, I. Molina-Fernandez, and R. Halir, “First monolithic InP-based 90°-hybrid OEIC comprising balanced detectors for 100GE coherent frontends,” in “IEEE International Conference on Indium Phosphide Related Materials (IPRM),” (2009), pp. 167–170.

Weinert, C.-M.

M. Seimetz and C.-M. Weinert, “Options, Feasibility, and Availability of 2×4 90° Hybrids for Coherent Optical Systems,” J. Lightw. Technol. 24, 1317 (2006).
[Crossref]

Xu, D.

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Xu, L.

L. Xu, X. Leijtens, B. Docter, T. de Vries, E. Smalbrugge, F. Karouta, and M. Smit, “MMI-reflector: A novel on-chip reflector for photonic integrated circuits,” in “European Conference on Optical Communications (ECOC),” (2009), pp. 1–2.

Xu, X.

Yang, J.

F. Wang, J. Yang, L. Chen, X. Jiang, and M. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. 18, 421–423 (2006).
[Crossref]

Ylinen, S.

Zavargo-Peche, L.

A. Maese-Novo, R. Halir, S. Romero-García, D. Pérez-Galacho, L. Zavargo-Peche, A. O.-M. nux, I. Molina-Fernández, J. G. Wangüemert-Pérez, and P. Cheben, “Wavelength independent multimode interference coupler,” Opt. Express 21, 7033–7040 (2013).
[Crossref] [PubMed]

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Zhang, Y.

Appl. Opt. (1)

Computer Physics Communications (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (1)

J. Azana and M. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Topics Quantum Electron. 7, 728–744 (2001).
[Crossref]

IEEE Photon. Technol. Lett. (5)

T. Fukuda, K. Okamoto, Y. Hinokuma, and K. Hamamoto, “Phase-Locked Array Laser Diodes (LDs) by Using 1xN Active Multimode-Interferometer (MMI),” IEEE Photon. Technol. Lett. 21, 176–178 (2009).
[Crossref]

H. Chen and A. Poon, “Low-Loss Multimode-Interference-Based Crossings for Silicon Wire Waveguides,” IEEE Photon. Technol. Lett. 18, 2260–2262 (2006).
[Crossref]

F. Wang, J. Yang, L. Chen, X. Jiang, and M. Wang, “Optical switch based on multimode interference coupler,” IEEE Photon. Technol. Lett. 18, 421–423 (2006).
[Crossref]

E. Pennings, R. van Roijen, M. van Stralen, P. de Waard, R. Koumans, and B. Verbeck, “Reflection properties of multimode interference devices,” IEEE Photon. Technol. Lett. 6, 715–718 (1994).
[Crossref]

E. Kleijn, D. Melati, A. Melloni, T. de Vries, M. Smit, and X. Leijtens, “Multimode Interference Couplers With Reduced Parasitic Reflections,” IEEE Photon. Technol. Lett. 26, 408–410 (2014).
[Crossref]

J. Lightw. Technol. (6)

E. Kleijn, M. Smit, and X. Leijtens, “Multimode Interference Reflectors: A New Class of Components for Photonic Integrated Circuits,” J. Lightw. Technol. 31, 3055–3063 (2013).
[Crossref]

M. Seimetz and C.-M. Weinert, “Options, Feasibility, and Availability of 2×4 90° Hybrids for Coherent Optical Systems,” J. Lightw. Technol. 24, 1317 (2006).
[Crossref]

P. Besse, M. Bachmann, H. Melchior, L. Soldano, and M. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightw. Technol. 12, 1004–1009 (1994).
[Crossref]

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

P. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightw. Technol. 14, 2286–2293 (1996).
[Crossref]

J. D. Doménech, J. S. Fandiño, B. Gargallo, and P. Muñoz, “Arbitrary Coupling Ratio Multimode Interference Couplers in Silicon-on-Insulator,” J. Lightw. Technol. 32, 2536–2543 (2014).
[Crossref]

J. Opt. Pure Appl. Opt. (1)

K. Solehmainen, T. Aalto, J. Dekker, M. Kapulainen, M. Harjanne, and P. Heimala, “Development of multi-step processing in Silicon-on-Insulator for optical waveguide applications,” J. Opt. Pure Appl. Opt. 8, S455–S460 (2006).
[Crossref]

Journal of Modern Optics (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” Journal of Modern Optics 43, 2139–2164 (1996).
[Crossref]

Laser & Photonics Reviews (1)

A. Ortega-Moux, C. Alonso-Ramos, A. Maese-Novo, R. Halir, L. Zavargo-Peche, D. Prez-Galacho, I. Molina-Fernndez, J. G. Wangemert-Prez, P. Cheben, J. H. Schmid, J. Lapointe, D. Xu, and S. Janz, “An ultra-compact multimode interference coupler with a subwavelength grating slot,” Laser & Photonics Reviews 7, L12–L15 (2013).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Photon. Res. (1)

Phys. Rev. A (1)

A. E. Kaplan, I. Marzoli, W. E. Lamb, and W. P. Schleich, “Multimode interference: Highly regular pattern formation in quantum wave-packet evolution,” Phys. Rev. A 61, 032101 (2000).
[Crossref]

Other (5)

L. Xu, X. Leijtens, B. Docter, T. de Vries, E. Smalbrugge, F. Karouta, and M. Smit, “MMI-reflector: A novel on-chip reflector for photonic integrated circuits,” in “European Conference on Optical Communications (ECOC),” (2009), pp. 1–2.

R. Kunkel, H.-G. Bach, D. Hoffmann, C. Weinert, I. Molina-Fernandez, and R. Halir, “First monolithic InP-based 90°-hybrid OEIC comprising balanced detectors for 100GE coherent frontends,” in “IEEE International Conference on Indium Phosphide Related Materials (IPRM),” (2009), pp. 167–170.

K. Takiguchi, T. Kitoh, M. Oguma, Y. Hashizume, and H. Takahashi, “Integrated-optic OFDM Demultiplexer using Multi-mode Interference Coupler-based Optical DFT Circuit,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2012), p. OM3J.6.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd Edition (Wiley, 2007), chap. 8, pp. 299–308.

S. J. Orfanidis, Electromagnetic Waves and Antennas (Rutgers University, 2008), chap. 8, pp. 305–307.

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Figures (9)

Fig. 1
Fig. 1 Example of how an MIR is formed. (a) Original 2×4 MMI. (b) Equivalent 4-port MIR after introduction of an aluminium mirror.
Fig. 2
Fig. 2 (a) and (b) 2×2 butterly MMIs for implementing arbitrary power splitting ratios, without and with crosscoupler, respectively. (c) and (d) Equivalent 2-port MIRs.
Fig. 3
Fig. 3 (a) Cross-section of the aluminium mirror layer stack. (b) 3D Model of a 2-port MIR with an aluminium mirror. (c) SEM picture of a fabricated device.
Fig. 4
Fig. 4 (a) Equivalent infinite slab waveguide after reduction of the original MIR cross-section with the effective index method. Each mode (n = 0, 1…) is represented here as a plane wave bouncing back and forth at the waveguide interfaces with a certain angle ( θ i n ). (b) Schematic representation of the resonant reflection of a plane wave (mode) upon incidence on the multilayer structure that forms the aluminium mirror.
Fig. 5
Fig. 5 (a) Simulated reflectivities for the type B device of Table 2 (ρ/τ = 0/100), both using FDTD (grey solid line) and the quasi-analytical theory (black dashed line). (b) Idem for the type A device of Table 2 (ρ/τ = 50/50).
Fig. 6
Fig. 6 (a) Schematic diagram of the structures employed to measure the effective reflectivities of the devices under test. (b) SEM picture of a test structure.
Fig. 7
Fig. 7 (a) Measured spectrum of the type A device of Table 2 (ρ/τ = 50/50), after normalization with a test straight waveguide. (b) Simulated spectrum of the same structure using the theoretical model of Section 4. (c) Measured transmission spectrum of a reference straight waveguide. (d) Relative contributions of the different resonant cavities present in the structure (see Fig. 6(a)).
Fig. 8
Fig. 8 (a) Measured spectrum of the type A device of Table 2 (ρ/τ = 50/50), after normalization with a test straight waveguide and smoothed with a 30 point moving average (≃120 pm). Maxima and minima are also shown as black triangles. (b) Extracted extinction ratios for different wavelengths. (c) Measured reflectivity and third-order polynomial fit (solid black line). (d) Reduction in reflectivity for each device in Table 2, as the smoothing window goes from zero to a size equal to the FSR of the FP.
Fig. 9
Fig. 9 (a) to (e) Average reflectivities (solid black lines) and standard deviations (grey dashed lines) for the 5 MIRs shown in Table 2, computed after measuring 9 different dies. (f) Black diamonds: Measured average and standard deviations of the reflectivities versus the simulated values at 1550 nm. Grey dashed line: Slope fitted to the data that provides an estimate of the average intrinsic losses both in the mirror and in the access waveguides.

Tables (3)

Tables Icon

Table 1 Parameters of 2-port MIRs. Note that if a crosscoupler is added, then the reflection/transmission ratios change for types A to C. The crosscoupler lengths and their corresponding ratios are separated by a vertical bar (|), where those with a crosscoupler have been highlighted in italics. Type D only works after the introduction of a crosscoupler, as explained in the text. Lπ is defined as λ 0 / ( 2 ( n ¯ 0 n ¯ 1 ) ) [ 2 ], where n ¯ 0 and n ¯ 1 are the effective refractive indices of the fundamental and first order modes of the multimode section, respectively. Input positions are measured with respect to the center of the MIR.

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Table 2 Physical dimensions and BPM simulation results of the 5 different fabricated devices. The width is common for all designs (11.7 µm), and no crosscouplers are employed (Lc/2=0).

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Table 3 Estimated reduction in measured reflectivity due to smoothing (See Fig. 8(d)).

Equations (6)

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θ i n = arccos ( β n | k n | ) = arccos ( n e f f n n e f f W G )
ρ n = ρ 12 n + ρ 23 n exp ( j 4 π ( d / λ 0 ) n e f f B G cos θ t n ) 1 + ρ 12 n + ρ 23 n exp ( j 4 π ( d / λ 0 ) n e f f B G cos θ t n )
θ t n = arcsin ( n e f f W G sin θ i n n e f f B G )
ρ 12 n = ( n e f f B G ) 2 n e f f W G cos θ i n ( n e f f W G ) 2 n e f f B G cos θ t n ( n e f f B G ) 2 n e f f W G cos θ i n + ( n e f f W G ) 2 n e f f B G cos θ t n
ρ 23 n = ( n A l ) 2 n e f f B G cos θ i n ( n e f f B G ) 2 [ ( n A l ) 2 ( n e f f B G cos θ t n ) 2 ] 1 / 2 ( n A l ) 2 n e f f B G cos θ i n + ( n e f f B G ) 2 [ ( n A l ) 2 ( n e f f B G cos θ t n ) 2 ] 1 / 2
ρ e f f = ER 1 ER + 1

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