Abstract

Integrated optic beam combiners offer many advantages over conventional bulk optic implementations for astronomical imaging. To our knowledge, integrated optic beam combiners have only been demonstrated at operating wavelengths below 4μm. Operation in the midinfrared wavelength region, however, is highly desirable. In this paper, a theoretical design technique based on three coupled waveguides is developed to achieve fully achromatic, broadband, polarization-insensitive, lossless beam combining. This design may make it possible to achieve the very deep broadband nulls needed for exoplanet searching.

© 2010 Optical Society of America

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2010

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “Integrated optic beam combiners for stellar interferometry and nulling at near- and mid-infrared wavelengths,” Proc. SPIE 7734, 77342T (2010).
[CrossRef]

2009

2007

2006

N. Ho, M. C. Phillips, H. Qiao, P. J. Allen, K. Krishnaswami, B. J. Riley, T. L. Meyers, and N. C. Anheier, Jr., “Single-mode low-loss chalcogenide glass waveguides for mid-infrared,” Opt. Lett. 31, 1860–1862 (2006).
[CrossRef] [PubMed]

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguide components for the long-wave infrared region,” J. Opt. A Pure Appl. Opt. 8, 840–848 (2006).
[CrossRef]

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

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

2002

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

2001

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

2000

1999

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

1992

A. Takagi, K. Jinguji, and M. Kawachi, “Design and fabrication of broad-band silica-based optical waveguide couplers with asymmetric structure,” J. Quantum Electron. 28, 848–854 (1992).
[CrossRef]

1990

K. Jinguji, N. Takato, A. Sugita, and M. Kawachi, “Mach-Zehnder interferometer type optical waveguide coupler with wavelength-flattened coupling ratio,” Electron. Lett. 26, 1326–1327 (1990).
[CrossRef]

1987

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

H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

1980

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
[CrossRef]

1975

Agarwal, A.

Allen, P. J.

Anheier, N. C.

Benech, P.

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

P. Haguenauer, J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics for astronomical interferometry. III. Optical validation of a planar optics two-telescope beam combiner,” Appl. Opt. 39, 2130–2139 (2000).
[CrossRef]

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics single-mode interferometric beam combiner for near infrared astronomy,” in Integrated Optics for Astronomical Interferometry (Bastianelli-Guirimand, 1996), pp. 195–204.

Bennett, B. R.

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

Berger, J.

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

Berger, J. P.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

P. Haguenauer, J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics for astronomical interferometry. III. Optical validation of a planar optics two-telescope beam combiner,” Appl. Opt. 39, 2130–2139 (2000).
[CrossRef]

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

Berger, J.-P.

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “Integrated optic beam combiners for stellar interferometry and nulling at near- and mid-infrared wavelengths,” Proc. SPIE 7734, 77342T (2010).
[CrossRef]

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “An infrared integrated optic astronomical beam combiner for stellar interferometry at 3–4μm,” Opt. Express 17, 18489–18500 (2009).
[CrossRef]

Buchwald, W. R.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguide components for the long-wave infrared region,” J. Opt. A Pure Appl. Opt. 8, 840–848 (2006).
[CrossRef]

Burns, W. K.

Carleton, N. P.

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Carlie, N.

Delboulbé, A.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

Emelett, S. J.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguide components for the long-wave infrared region,” J. Opt. A Pure Appl. Opt. 8, 840–848 (2006).
[CrossRef]

Feng, N.-N.

Foresto, V. C. d.

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Glindemann, A.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

Gluck, S.

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

Haguenauer, P.

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

P. Haguenauer, J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics for astronomical interferometry. III. Optical validation of a planar optics two-telescope beam combiner,” Appl. Opt. 39, 2130–2139 (2000).
[CrossRef]

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

Hattori, H. T.

V. M. Schneider and H. T. Hattori, “High-tolerance power splitting in symmetric triple-mode evolution couplers,” IEEE J. Quantum Electron. 36, 923–930 (2000).
[CrossRef]

Haus, H.

H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Ho, N.

Hsiao, H.-K.

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “Integrated optic beam combiners for stellar interferometry and nulling at near- and mid-infrared wavelengths,” Proc. SPIE 7734, 77342T (2010).
[CrossRef]

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “An infrared integrated optic astronomical beam combiner for stellar interferometry at 3–4μm,” Opt. Express 17, 18489–18500 (2009).
[CrossRef]

H.-K. Hsiao, K. A. Winick, and J. D. Monnier, “A mid-infrared integrated optic astronomical beam combiner for stellar interferometry,” in OSA Frontiers in Optics (FiO) (OSA, 2009), paper FMJ1.

Hu, J.

Huang, W.

H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Ishikawa, H.

Jinguji, K.

A. Takagi, K. Jinguji, and M. Kawachi, “Design and fabrication of broad-band silica-based optical waveguide couplers with asymmetric structure,” J. Quantum Electron. 28, 848–854 (1992).
[CrossRef]

K. Jinguji, N. Takato, A. Sugita, and M. Kawachi, “Mach-Zehnder interferometer type optical waveguide coupler with wavelength-flattened coupling ratio,” Electron. Lett. 26, 1326–1327 (1990).
[CrossRef]

Jocou, L.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

Kawachi, M.

A. Takagi, K. Jinguji, and M. Kawachi, “Design and fabrication of broad-band silica-based optical waveguide couplers with asymmetric structure,” J. Quantum Electron. 28, 848–854 (1992).
[CrossRef]

K. Jinguji, N. Takato, A. Sugita, and M. Kawachi, “Mach-Zehnder interferometer type optical waveguide coupler with wavelength-flattened coupling ratio,” Electron. Lett. 26, 1326–1327 (1990).
[CrossRef]

Kawakami, S.

H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Kern, P.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

P. Haguenauer, J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics for astronomical interferometry. III. Optical validation of a planar optics two-telescope beam combiner,” Appl. Opt. 39, 2130–2139 (2000).
[CrossRef]

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics single-mode interferometric beam combiner for near infrared astronomy,” in Integrated Optics for Astronomical Interferometry (Bastianelli-Guirimand, 1996), pp. 195–204.

Kimerling, L.

Krishnaswami, K.

Labeye, P.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

Labeyrie, A.

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge U. Press, 2006).
[CrossRef]

Lacasse, M. G.

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Laurent, E.

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

LeBouquin, J. B.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

Li, H. H.

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
[CrossRef]

Lipson, S. G.

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge U. Press, 2006).
[CrossRef]

Malbet, F.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

P. Haguenauer, J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics for astronomical interferometry. III. Optical validation of a planar optics two-telescope beam combiner,” Appl. Opt. 39, 2130–2139 (2000).
[CrossRef]

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics single-mode interferometric beam combiner for near infrared astronomy,” in Integrated Optics for Astronomical Interferometry (Bastianelli-Guirimand, 1996), pp. 195–204.

Mariotti, J. M.

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Mazé, G.

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Mennesson, B.

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Meyers, T. L.

Millan-Gabet, R.

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

Milton, A. F.

Monnier, J. D.

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “Integrated optic beam combiners for stellar interferometry and nulling at near- and mid-infrared wavelengths,” Proc. SPIE 7734, 77342T (2010).
[CrossRef]

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “An infrared integrated optic astronomical beam combiner for stellar interferometry at 3–4μm,” Opt. Express 17, 18489–18500 (2009).
[CrossRef]

H.-K. Hsiao, K. A. Winick, and J. D. Monnier, “A mid-infrared integrated optic astronomical beam combiner for stellar interferometry,” in OSA Frontiers in Optics (FiO) (OSA, 2009), paper FMJ1.

Nayfeh, A.

A. Nayfeh, “Heteroepitaxial growth of relaxed germanium on silicon,” Ph.D. thesis (Stanford University, 2006).

Nisenson, P.

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge U. Press, 2006).
[CrossRef]

Perrin, G.

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Petit, L.

Phillips, M. C.

Qiao, H.

Reynaud, F.

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

Richardson, K.

Ridgway, S.

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Riley, B. J.

Rousselet-Perraut, K.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

P. Haguenauer, J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics for astronomical interferometry. III. Optical validation of a planar optics two-telescope beam combiner,” Appl. Opt. 39, 2130–2139 (2000).
[CrossRef]

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

Schanen, I.

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

Schanen-Duport, I.

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

P. Haguenauer, J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics for astronomical interferometry. III. Optical validation of a planar optics two-telescope beam combiner,” Appl. Opt. 39, 2130–2139 (2000).
[CrossRef]

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics single-mode interferometric beam combiner for near infrared astronomy,” in Integrated Optics for Astronomical Interferometry (Bastianelli-Guirimand, 1996), pp. 195–204.

Schneider, V. M.

V. M. Schneider and H. T. Hattori, “High-tolerance power splitting in symmetric triple-mode evolution couplers,” IEEE J. Quantum Electron. 36, 923–930 (2000).
[CrossRef]

Schöeller, M.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

Severi, M.

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

Soref, R. A.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguide components for the long-wave infrared region,” J. Opt. A Pure Appl. Opt. 8, 840–848 (2006).
[CrossRef]

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

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

Sugita, A.

K. Jinguji, N. Takato, A. Sugita, and M. Kawachi, “Mach-Zehnder interferometer type optical waveguide coupler with wavelength-flattened coupling ratio,” Electron. Lett. 26, 1326–1327 (1990).
[CrossRef]

Takagi, A.

A. Takagi, K. Jinguji, and M. Kawachi, “Design and fabrication of broad-band silica-based optical waveguide couplers with asymmetric structure,” J. Quantum Electron. 28, 848–854 (1992).
[CrossRef]

Takato, N.

K. Jinguji, N. Takato, A. Sugita, and M. Kawachi, “Mach-Zehnder interferometer type optical waveguide coupler with wavelength-flattened coupling ratio,” Electron. Lett. 26, 1326–1327 (1990).
[CrossRef]

Tamir, T.

T. Tamir, Guided-Wave Optoelectronics, 2nd ed. (Springer-Verlag, 1990), Chap. 3.
[CrossRef]

Tarasov, V.

Traub, W.

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

Traub, W. A.

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Whitaker, N.

H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Winick, K. A.

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “Integrated optic beam combiners for stellar interferometry and nulling at near- and mid-infrared wavelengths,” Proc. SPIE 7734, 77342T (2010).
[CrossRef]

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “An infrared integrated optic astronomical beam combiner for stellar interferometry at 3–4μm,” Opt. Express 17, 18489–18500 (2009).
[CrossRef]

H.-K. Hsiao, K. A. Winick, and J. D. Monnier, “A mid-infrared integrated optic astronomical beam combiner for stellar interferometry,” in OSA Frontiers in Optics (FiO) (OSA, 2009), paper FMJ1.

Zabihian, F.

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

J. Berger, P. Haguenauer, P. Kern, K. Rousselet-Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, and W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34(2001).
[CrossRef]

E. Laurent, K. Rousselet-Perraut, P. Benech, J. P. Berger, S. Gluck, P. Haguenauer, P. Kern, F. Malbet, and I. Schanen-Duport, “Integrated optics for astronomical interferometry. V. Extension to the K band,” Astron. Astrophys. 390, 1171–1176 (2002).
[CrossRef]

J. B. LeBouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J. P. Berger, A. Delboulbé, P. Kern, A. Glindemann, and M. Schöeller, “Integrated optics for astronomical interferometry. VI. Coupling the light of the VLTI in K band,” Astron. Astrophys. 450, 1259–1264 (2006).
[CrossRef]

B. Mennesson, J. M. Mariotti, V. C. d. Foresto, G. Perrin, S. Ridgway, W. A. Traub, N. P. Carleton, M. G. Lacasse, and G. Mazé, “Thermal infrared stellar interferometry using single-mode guided optics: first results with the TISIS experiment on IOTA,” Astron. Astrophys. 346, 181–189 (1999).

Astron. Astrophys. Suppl. Ser.

J. P. Berger, K. Rousselet-Perraut, P. Kern, F. Malbet, I. Schanen-Duport, F. Reynaud, P. Haguenauer, and P. Benech, “Integrated optics for astronomical interferometry. II. First laboratory white-light interferograms,” Astron. Astrophys. Suppl. Ser. 139, 173–177 (1999).
[CrossRef]

Electron. Lett.

K. Jinguji, N. Takato, A. Sugita, and M. Kawachi, “Mach-Zehnder interferometer type optical waveguide coupler with wavelength-flattened coupling ratio,” Electron. Lett. 26, 1326–1327 (1990).
[CrossRef]

IEEE J. Quantum Electron.

V. M. Schneider and H. T. Hattori, “High-tolerance power splitting in symmetric triple-mode evolution couplers,” IEEE J. Quantum Electron. 36, 923–930 (2000).
[CrossRef]

J. Lightwave Technol.

H. Ishikawa, “Fully adiabatic design of waveguide branches,” J. Lightwave Technol. 25, 1832–1840 (2007).
[CrossRef]

H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

J. Opt. A Pure Appl. Opt.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguide components for the long-wave infrared region,” J. Opt. A Pure Appl. Opt. 8, 840–848 (2006).
[CrossRef]

J. Phys. Chem. Ref. Data

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
[CrossRef]

J. Quantum Electron.

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

A. Takagi, K. Jinguji, and M. Kawachi, “Design and fabrication of broad-band silica-based optical waveguide couplers with asymmetric structure,” J. Quantum Electron. 28, 848–854 (1992).
[CrossRef]

J. Sel. Top. Quantum Electron.

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

Opt. Express

Opt. Lett.

Proc. SPIE

H.-K. Hsiao, K. A. Winick, J. D. Monnier, and J.-P. Berger, “Integrated optic beam combiners for stellar interferometry and nulling at near- and mid-infrared wavelengths,” Proc. SPIE 7734, 77342T (2010).
[CrossRef]

Other

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge U. Press, 2006).
[CrossRef]

P. Kern, F. Malbet, I. Schanen-Duport, and P. Benech, “Integrated optics single-mode interferometric beam combiner for near infrared astronomy,” in Integrated Optics for Astronomical Interferometry (Bastianelli-Guirimand, 1996), pp. 195–204.

H.-K. Hsiao, K. A. Winick, and J. D. Monnier, “A mid-infrared integrated optic astronomical beam combiner for stellar interferometry,” in OSA Frontiers in Optics (FiO) (OSA, 2009), paper FMJ1.

T. Tamir, Guided-Wave Optoelectronics, 2nd ed. (Springer-Verlag, 1990), Chap. 3.
[CrossRef]

A. Nayfeh, “Heteroepitaxial growth of relaxed germanium on silicon,” Ph.D. thesis (Stanford University, 2006).

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Figures (10)

Fig. 1
Fig. 1

Three coupled channel waveguides. Waveguides 1 and 3 are identical and equidistant from waveguide 2.

Fig. 2
Fig. 2

Schematic plot of proposed achromatic beam combiner.

Fig. 3
Fig. 3

Cross-sectional view of the coupled two- waveguide system along with refractive index values.

Fig. 4
Fig. 4

Proposed candidates for midinfrared waveguide fabrication: (a) silicon (or germanium) rib membrane waveguide; (b) Ge/Si heterostructure raised-strip waveguide.

Fig. 5
Fig. 5

(a) Schematic of broadband achromatic beam combiner. (b) Cross section of Ge/Si raised-strip waveguide geometry for fundamental mode calculation by beam propagation method.

Fig. 6
Fig. 6

(a) TE-mode profile at λ 0 = 10 μm , n eff = 3.6017 . (b) TM-mode profile at λ 0 = 10 μm , n eff = 3.6454 with nominal design ( H = W = 3.5 μm ).

Fig. 7
Fig. 7

Power transferred from the TE local normal mode Ψ + ( x , y ; z ) to the TE local normal mode Ψ ( x , y ; z ) at λ 0 = 10 μm .

Fig. 8
Fig. 8

(Left) Variation of the coupling coefficient, κ 12 ( z ) , and the dephasing term, Δ β ( z ) , along propagation direction z. (Right) Nonadiabatic term, ξ + ( z ) , for the Blackman function for the TE mode at λ 0 = 10 μm .

Fig. 9
Fig. 9

(Top) Total fraction of power remaining when only the local normal mode Ψ + is excited at z = 0 , i.e., a + ( 0 ) = 1 and a ( 0 ) = 0 . (Bottom) Fraction of launched power in Ψ + mode ( a + ( 0 ) = 1 and a ( 0 ) = 0 ) transferred to the local normal mode Ψ .

Fig. 10
Fig. 10

(a) Width variation of the outer waveguides as a function of propagation distance. (b) Gap variation between the outer and the central waveguides as a function of propagation distance.

Tables (4)

Tables Icon

Table 1 Device Parameters a

Tables Icon

Table 2 Dispersion of Si and Ge at 20 ° C and Different Midinfrared Wavelengths

Tables Icon

Table 3 Fractional Power Coupled into the TE and TM Ψ Mode at L = 6000 μm Evaluated at Different Wavelengths

Tables Icon

Table 4 Fractional Power Coupled to Other Mode, i.e., 1 a 2 2 ( L ) , Due to Imperfection of the Fabrication Process

Equations (98)

Equations on this page are rendered with MathJax. Learn more.

E ( x , y , z ) = Ψ ( x , y ) exp ( j β z ) = l = 1 3 a l ( z ) Φ l ( x , y ) ,
+ + Φ l 2 ( x , y ) d x d y = 1 , and l = 1 , 2 , 3 ,
d a 1 d z = j β 1 a 1 + j κ 12 a 2 + j κ 13 a 3 , d a 2 d z = j κ 12 a 1 + j β 2 a 2 + j κ 23 a 3 , d a 3 d z = j κ 13 a 1 + j κ 23 a 2 + j β 3 a 3 ,
E l ( x , y ; z ) = Ψ l ( x , y ; z ) exp ( j β l ( z ) z ) = a 1 ( l ) ( z ) Φ 1 ( x , y ; z ) + a 2 ( l ) ( z ) Φ 2 ( x , y ; z ) + a 3 ( l ) ( z ) Φ 3 ( x , y ; z ) ,
A l ( z ) = exp ( j β l ( z ) z ) ( a 1 ( l ) a 2 ( l ) a 3 ( l ) ) , l = + , , 0
Ψ l ( x , y ; z ) = ( Φ 1 ( x , y ; z ) Φ 2 ( x , y ; z ) Φ 3 ( x , y ; z ) ) A l ( z ) , l = + , , 0.
A + = ( e / 2 d e / 2 ) , A = ( d / 2 e d / 2 ) , A 0 = ( 1 / 2 0 1 / 2 ) ,
d = 1 2 ( 1 + X 1 + X 2 ) , e = 1 2 ( 1 X 1 + X 2 ) .
Ψ i | z | Ψ j + + Ψ i * ( x , y ; z ) z Ψ j ( x , y ; z ) d x d y .
| a 1 | exp ( j ϕ 1 ) ( 1 0 0 ) = | a 1 | exp ( j ϕ 1 ) [ 1 2 ( 1 0 1 ) + 1 2 ( 1 0 1 ) ] = | a 1 | exp ( j ϕ 1 ) [ 1 2 A 0 ( 0 ) + 1 2 A + ( 0 ) ] ,
| a 3 | exp ( j ϕ 3 ) ( 0 0 1 ) = | a 3 | exp ( j ϕ 3 ) [ 1 2 ( 1 0 1 ) + 1 2 ( 1 0 1 ) ] = | a 3 | exp ( j ϕ 3 ) [ 1 2 A 0 ( 0 ) + 1 2 A + ( 0 ) ] ,
( | a 1 | exp ( j ϕ 1 ) 0 | a 3 | exp ( j ϕ 3 ) ) = [ 1 2 | a 1 | exp ( j ϕ 1 ) 1 2 | a 3 | exp ( j ϕ 3 ) ] A 0 ( 0 ) + [ 1 2 | a 1 | exp ( j ϕ 1 ) + 1 2 | a 3 | exp ( j ϕ 3 ) ] A + ( 0 ) .
A 0 ( L ) = ( 1 2 0 1 2 ) exp ( j 0 L β 0 ( z ) d z ) ,
A + ( L ) = ( 0 1 0 ) exp ( j 0 L β + ( z ) d z ) ,
[ | a 1 | exp ( j ϕ 1 ) | a 3 | exp ( j ϕ 3 ) ] exp ( j 0 L β 0 ( z ) d z ) 1 2 ( 1 0 1 ) + [ | a 1 | exp ( j ϕ 1 ) + | a 3 | exp ( j ϕ 3 ) ] exp ( j 0 L β + ( z ) d z ) 1 2 ( 0 1 0 ) .
1 4 | | a 1 | exp ( j ϕ 1 ) | a 3 | exp ( j ϕ 3 ) | 2 = 1 4 ( | a 1 | 2 + | a 3 | 2 ) 1 2 | a 1 | | a 3 | cos ( ϕ 1 ϕ 3 ) : at output ports 1 and 3 ,
1 2 | | a 1 | exp ( j ϕ 1 ) + | a 3 | exp ( j ϕ 3 ) | 2 = 1 2 ( | a 1 | 2 + | a 3 | 2 ) + | a 1 | | a 3 | cos ( ϕ 1 ϕ 3 ) : at output port 2 .
( 2 + k 0 2 n 2 ) E + 2 ( ( ln n ) · E ) = 0 .
E i ( x , y ; z ) = Φ i ( x , y ; z ) exp ( j β 0 ( i ) z ) , i = x or y ,
j z Φ x = 1 2 n 0 k 0 ( Φ x x 2 + Φ x y 2 ) + k 0 ( n 2 ( x , y ; z ) n 0 2 ) 2 n 0 Φ x for TM ;
j z Φ y = 1 2 n 0 k 0 ( Φ y x 2 + Φ y y 2 ) + k 0 ( n 2 ( x , y ; z ) n 0 2 ) 2 n 0 Φ y for TE .
j z E i ( x , y ; z ) = B i ( z ) · E i ( x , y ; z ) ,
B x ( z ) = 1 2 n 0 k 0 ( x 2 + y 2 ) + k 0 ( n 2 ( x , y ; z ) + n 0 2 ) 2 n 0 for TM ;
B y ( z ) = 1 2 n 0 k 0 ( x 2 + y 2 ) + k 0 ( n 2 ( x , y ; z ) + n 0 2 ) 2 n 0 for TE .
B ( z ) Ψ l ( x , y ; z ) = β l ( z ) Ψ l ( x , y ; z ) , l = + , , 0 .
Ψ 0 ( x , y ; z ) | Ψ 0 ( x , y ; z ) = Ψ + ( x , y ; z ) | Ψ + ( x , y ; z ) = Ψ ( x , y ; z ) | Ψ ( x , y ; z ) = 1 ,
f ( x , y ; z ) | g ( x , y ; z ) = + + f * ( x , y ; z ) g ( x , y ; z ) d x d y .
Ψ l ( x , y ; z ) | z | Ψ l ( x , y ; z ) = 0 , l = 0 , + , .
Ψ l ( x , y ; z ) | z | Ψ k ( x , y ; z ) = 0 ,     l k ,     { l , k } = 0 , + , ;
Ψ l ( x , y ; z ) Ψ k ( x , y ; z ) = 0 ,     l k ,     { l , k } = 0 , + , .
E ( x , y , z ) = a 0 ( z ) Ψ 0 ( x , y ; z ) + a + ( z ) Ψ + ( x , y ; z ) + a ( z ) Ψ ( x , y ; z ) .
j z E ( x , y , z ) = B ( z ) E ( x , y , z )
j a 0 ( z ) z Ψ 0 ( x , y ; z ) j a 0 ( z ) Ψ 0 ( z ) z j a + ( z ) z Ψ + ( x , y ; z ) j a + ( z ) Ψ + ( z ) z j a ( z ) z Ψ ( x , y ; z ) j a ( z ) Ψ ( z ) z = a 0 ( z ) B ( z ) Ψ 0 ( x , y ; z ) + a + ( z ) B ( z ) Ψ + ( x , y ; z ) + a ( z ) B ( z ) Ψ ( x , y ; z ) = a 0 ( z ) β 0 ( z ) Ψ 0 ( x , y ; z ) + a + ( z ) β + ( z ) Ψ + ( x , y ; z ) + a ( z ) β ( z ) Ψ ( x , y ; z ) ,
B ( z ) Ψ 0 ( x , y ; z ) = β 0 ( z ) Ψ 0 ( x , y ; z ) ,
B ( z ) Ψ + ( x , y ; z ) = β + ( z ) Ψ + ( x , y ; z ) ,
B ( z ) Ψ ( x , y ; z ) = β ( z ) Ψ ( x , y ; z ) .
j a + ( z ) z j a 0 ( z ) Ψ + ( x , y ; z ) | z | Ψ 0 ( x , y ; z ) j a ( z ) Ψ + ( x , y ; z ) | z | Ψ ( x , y ; z ) = a + ( z ) β + ( z ) .
j a 0 ( z ) z j a + ( z ) Ψ 0 ( x , y ; z ) | z | Ψ + ( x , y ; z ) j a ( z ) Ψ 0 ( x , y ; z ) | z | Ψ ( x , y ; z ) = a 0 ( z ) β 0 ( z ) ,
j a ( z ) z j a 0 ( z ) Ψ ( x , y ; z ) | z | Ψ 0 ( x , y ; z ) j a + ( z ) Ψ ( x , y ; z ) | z | Ψ + ( x , y ; z ) = a ( z ) β ( z ) .
j z ( a 0 ( z ) a + ( z ) a ( z ) ) = ( β 0 ( z ) + j ξ 0 + ( z ) + j ξ 0 ( z ) j ξ 0 + ( z ) β + ( z ) + j ξ + ( z ) j ξ 0 ( z ) j ξ + ( z ) β ( z ) ) · ( a 0 ( z ) a + ( z ) a ( z ) ) ,
ξ l k ( z ) Ψ l ( x , y ; z ) | z | Ψ k ( x , y ; z ) , l k , { l , k } = 0 , + , .
z Ψ l ( x , y ; z ) = 0 or 0 , l = 0 , + , .
Ψ + ( x , y ; z ) = a 1 ( + ) ( z ) Φ 1 ( x , y ; z ) + a 2 ( + ) ( z ) Φ 2 ( x , y ; z ) + a 1 ( + ) ( z ) Φ 3 ( x , y ; z ) , Ψ ( x , y ; z ) = a 1 ( ) ( z ) Φ 1 ( x , y ; z ) + a 2 ( ) ( z ) Φ 2 ( x , y ; z ) + a 1 ( ) ( z ) Φ 3 ( x , y ; z ) , Ψ 0 ( x , y ; z ) = a 1 ( 0 ) ( z ) Φ 1 ( x , y ; z ) a 1 ( 0 ) ( z ) Φ 3 ( x , y ; z ) ,
Φ l ( x , y ; z ) | Φ l ( x , y ; z ) = 1 , l = 1 , 2 , 3 .
Φ l ( x , y ; z ) | z | Φ l ( x , y ; z ) = 0 , l = 1 , 2 , 3 .
Φ 1 ( x , y ; z ) | z | Φ 1 ( x , y ; z ) = Φ 3 ( x , y ; z ) | z | Φ 3 ( x , y ; z ) , Φ 1 ( x , y ; z ) | z | Φ 3 ( x , y ; z ) = Φ 3 ( x , y ; z ) | z | Φ 1 ( x , y ; z ) , Φ 2 ( x , y ; z ) | Φ 1 ( x , y ; z ) = Φ 2 ( x , y ; z ) | Φ 3 ( x , y ; z ) .
ξ 0 + ( z ) = Ψ 0 ( x , y ; z ) | z | Ψ + ( x , y ; z ) = 0 , ξ 0 ( z ) = Ψ 0 ( x , y ; z ) | z | Ψ ( x , y ; z ) = 0 .
j z ( a 0 ( z ) a + ( z ) a ( z ) ) = ( β 0 ( z ) 0 0 0 β + ( z ) + j ξ + ( z ) 0 j ξ + ( z ) β ( z ) ) · ( a 0 ( z ) a + ( z ) a ( z ) ) .
n 1 + 2 + 3 2 ( x , y ; z ) = n cl 2 + Δ n 1 2 ( x , y ; z ) + Δ n 2 2 ( x , y ; z ) + Δ n 3 2 ( x , y ; z ) : coupled three-waveguide ;
n 1 + 3 2 ( x , y ; z ) = n cl 2 + Δ n 1 2 ( x , y ; z ) + Δ n 3 2 ( x , y ; z ) : waveguides 1 and 3 ;
n 2 2 ( x , y ; z ) = n cl 2 + Δ n 2 2 ( x , y ; z ) : waveguide 2 alone ;
Δ n i 2 ( x , y ; z ) = n core 2 ( x , y ; z ) n cl 2 :     inside cores , = 0   otherwise ,
B 1 + 2 + 3 ( z ) · Ψ q ( x , y ; z ) = β q ( z ) · Ψ q ( x , y ; z ) ,
B 1 + 3 ( z ) · Ψ 1 ( x , y ; z ) = β 1 + 3 ( z ) · Ψ 1 ( x , y ; z ) ,
B 2 ( z ) · Ψ 2 ( x , y ; z ) = β 2 ( z ) · Ψ 2 ( x , y ; z ) ,
( Ψ + ( x , y ; z ) | Ψ + ( x , y ; z ) Ψ + ( x , y ; z ) | Ψ ( x , y ; z ) Ψ ( x , y ; z ) | Ψ + ( x , y ; z ) Ψ ( x , y ; z ) | Ψ ( x , y ; z ) ) = ( 1 0 0 1 ) .
Ψ 1 ( x , y ; z ) = 1 2 [ Φ 1 ( x , y ; z ) + Φ 3 ( x , y ; z ) ] ,
Ψ 2 ( x , y ; z ) = Φ 2 ( x , y ; z ) ,
( Ψ 1 ( x , y ; z ) | Ψ 1 ( x , y ; z ) Ψ 1 ( x , y ; z ) | Ψ 2 ( x , y ; z ) Ψ 2 ( x , y ; z ) | Ψ 1 ( x , y ; z ) Ψ 2 ( x , y ; z ) | Ψ 2 ( x , y ; z ) ) = ( 1 S ( z ) S ( z ) 1 ) .
( Ψ + ( x , y ; z ) Ψ ( x , y ; z ) ) t = ( Ψ 1 ( x , y ; z ) Ψ 2 ( x , y ; z ) ) t · ( c 1 + ( z ) c 1 ( z ) c 2 + ( z ) c 2 ( z ) ) .
( B 11 ( z ) B 12 ( z ) B 21 ( z ) B 22 ( z ) ) · ( c 1 + ( z ) c 1 ( z ) c 2 + ( z ) c 2 ( z ) ) = ( 1 S ( z ) S ( z ) 1 ) · ( c 1 + ( z ) c 1 ( z ) c 2 + ( z ) c 2 ( z ) ) · ( β + ( z ) 0 0 β ( z ) ) ,
( B 11 ( z ) B 12 ( z ) B 21 ( z ) B 22 ( z ) ) = ( 1 S ( z ) S ( z ) 1 ) · ( β 1 ( z ) 0 0 β 2 ( z ) ) + ( κ 11 ( z ) κ 12 ( z ) κ 21 ( z ) κ 22 ( z ) ) ,
κ i j ( z ) = k 0 2 n 0 Ψ i | Δ N ( 3 j ) | Ψ j Ψ j | Ψ j ,
Δ N ( 1 ) = Δ n 1 2 ( x , y ; z ) + Δ n 3 2 ( x , y ; z ) ,
Δ N ( 2 ) = Δ n 2 2 ( x , y ; z ) .
( c 1 + ( z ) c 1 ( z ) c 2 + ( z ) c 2 ( z ) ) t ( 1 S ( z ) S ( z ) 1 ) ( c 1 + ( z ) c 1 ( z ) c 2 + ( z ) c 2 ( z ) ) = ( 1 0 0 1 ) .
tan θ ( z ) = κ 12 ( z ) δ 12 ( z ) ,
tan ϕ ( z ) = S ( z ) 1 S ( z ) 2 ,
δ 12 ( z ) = β 1 + κ 11 ( β 2 + κ 22 ) 2 1 S 2 ,
κ 12 ( z ) = κ 12 + κ 21 2 ( 1 S 2 ) S 1 S 2 κ 11 + κ 22 2 ,
( c 1 + ( z ) c 1 ( z ) c 2 + ( z ) c 2 ( z ) ) = 1 cos ϕ ( z ) ( cos θ ( z ) + ϕ ( z ) 2 sin θ ( z ) + ϕ ( z ) 2 sin θ ( z ) ϕ ( z ) 2 cos θ ( z ) ϕ ( z ) 2 ) ,
β ± ( z ) = β ¯ ( z ) ± δ 12 ( z ) 2 + κ 12 ( z ) 2 ,
β ¯ ( z ) = β 1 + κ 11 + β 2 + κ 22 2 ( 1 S 2 ) S 2 ( 1 S 2 ) ( κ 12 + κ 21 + S β 1 + S β 2 ) .
ξ + ( z ) = 1 2 θ ( z ) z cos θ ( z ) 2 cos ϕ ( z ) ϕ ( z ) z 1 cos ϕ ( z ) Ψ 2 ( x , y ; z ) | z | Ψ 1 ( x , y ; z ) .
z Ψ 2 ( x , y ; z ) = 0 ,
ξ + ( z ) = 1 2 θ ( z ) z cos θ ( z ) 2 cos ϕ ( z ) ϕ ( z ) z .
ξ + ( z ) 1 2 θ ( z ) z .
tan θ ( z ) = κ 12 ( z ) δ 12 ( z ) ,
δ 12 ( z ) β 1 ( z ) + κ 11 ( z ) ( β 2 ( z ) + κ 22 ( z ) ) 2 ,
κ 12 ( z ) κ 12 ( z ) + κ 21 ( z ) 2 .
κ 11 ( z ) = κ 22 ( z ) ,
κ 12 ( z ) = κ 21 ( z ) ,
κ 11 ( z ) κ 22 ( z ) ,
κ 12 ( z ) κ 21 ( z ) .
δ 12 ( z ) β 1 ( z ) β 2 ( z ) 2 = Δ β ( z ) 2 ,
κ 12 ( z ) κ 12 ( z ) ,
tan θ ( z ) 2 κ 12 ( z ) Δ β ( z ) .
d a d z = j β a a + j K b , d b d z = j K a + j β b b ,
a ( z ) = a 2 ( z ) , b ( z ) = 2 a 1 ( z ) = 2 a 3 ( z ) .
K ( z ) = K max sin ϑ ( z ) ,
Δ β ( z ) = Δ β max cos ϑ ( z ) ,
ξ + ( z ) 1 2 θ ( z ) z ,
ξ + ( z ) 1 2 ( 1 + X 2 ) X z .
ϑ ( z ) = π z L : Linear Function ;
ϑ ( z ) = π z L 0.5 sin 2 π z L : Raised Cosine Function ;
ϑ ( z ) = π z L 0.426 sin 2 π z L : Hamming Function ;
ϑ ( z ) = π z L 0.5952 sin 2 π z L + 0.0476 sin 4 π z L : Blackman Function .
δ = κ 23 ( z ) κ 12 ( z ) κ 12 ( z ) or δ = β 3 ( z ) β 1 ( z ) β 1 ( z ) ,

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