Abstract

Integrated-optic, astronomical, two-beam and three-beam, interferometric combiners have been designed and fabricated for operation in the L band (3 μm – 4 μm) for the first time. The devices have been realized in titanium-indiffused, x-cut lithium niobate substrates, and on-chip electro-optic fringe scanning has been demonstrated. White light fringes were produced in the laboratory using the two-beam combiner integrated with an on-chip Y-splitter.

© 2009 OSA

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    [CrossRef]

2009 (4)

2008 (2)

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

S. Brustlein, L. Del Rio, A. Tonello, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Laboratory demonstration of an infrared-to-visible up-conversion interferometer for spatial coherence analysis,” Phys. Rev. Lett. 100(15), 153903 (2008).
[CrossRef] [PubMed]

2006 (5)

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(3), 1259–1264 (2006).
[CrossRef]

G. Li, T. Eckhause, K. A. Winick, J. D. Monnier, and J. P. Berger, “Integrated optic beam combiners in lithium niobate for stellar interferometer,” Proc. SPIE 6268, 626834 (2006).
[CrossRef]

N. Hô, M. C. Phillips, H. Qiao, P. J. Allen, K. Krishnaswami, B. J. Riley, T. L. Myers, and N. C. Anheier., “Single-mode low-loss chalcogenide glass waveguides for the mid-infrared,” Opt. Lett. 31(12), 1860–1862 (2006).
[CrossRef] [PubMed]

Y. Sakamaki, T. Saida, M. Tamura, T. Hashimoto, and H. Takahashi, “Low loss crosstalk waveguide crossings designed by wavefront matching method,” IEEE Photon. Technol. Lett. 18(19), 2005–2007 (2006).
[CrossRef]

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

2005 (1)

J. Breckinridge and C. Lindensmith, “The astronomical search for origins,” Opt. Photonics News 16(2), 24–29 (2005).
[CrossRef]

2004 (1)

C. V. M. Fridlund, “The Darwin mission,” Adv. Space Res. 34(3), 613–617 (2004).
[CrossRef]

2002 (1)

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(3), 1171–1176 (2002).
[CrossRef]

2001 (3)

K. Kishioka, “A design method to achieve wide wavelength-flattened responses in directional coupler-type optical power splitters,” J. Lightwave Technol. 19(11), 1705–1715 (2001).
[CrossRef]

J. P. 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]

J. D. Monnier, “Asymmetric beam combination for optical interferometry,” Publ. Astron. Soc. Pac. 113(783), 639–645 (2001).
[CrossRef]

2000 (1)

1999 (2)

B. Mennesson, J. M. Mariotti, V. Coudé du 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(1), 173–177 (1999).
[CrossRef]

1988 (1)

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with the scalar finite-element method,” J. Lightwave Technol. 6(6), 1126–1135 (1988).
[CrossRef]

1985 (2)

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3(2), 385–391 (1985).
[CrossRef]

R. A. Becker, R. H. Rediker, and T. A. Lind, “Wide-bandwidth guided-wave electro-optic intensity modulator at λ = 3.39 μm,” Appl. Phys. Lett. 46(9), 809–811 (1985).
[CrossRef]

1982 (1)

W. J. Minford, S. K. Korotky, and R. C. Alferness, “Low-loss Ti:LiNbO3 waveguide bends at λ = 1.3 μm,” IEEE J. Quantum Electron. QE-18, 1802–1806 (1982).
[CrossRef]

1979 (1)

1978 (1)

R. N. Bracewell, “Detecting nonsolar planets by spinning infrared interferometers,” Nature 274(5673), 780–781 (1978).
[CrossRef]

1891 (1)

A. A. Michelson, “Measurement of Jupiter’s satellites by interference,” Nature 45(1155), 160–161 (1891).
[CrossRef]

Abel-Tiberini, L.

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Alferness, R. C.

W. J. Minford, S. K. Korotky, and R. C. Alferness, “Low-loss Ti:LiNbO3 waveguide bends at λ = 1.3 μm,” IEEE J. Quantum Electron. QE-18, 1802–1806 (1982).
[CrossRef]

R. C. Alferness, R. V. Schmidt, and E. H. Turner, “Characteristics of Ti-diffused lithium niobate optical directional couplers,” Appl. Opt. 18(23), 4012–4016 (1979).
[CrossRef] [PubMed]

Allen, P. J.

Anheier, N. C.

Arezki, B.

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Barillot, M.

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Bava, G. P.

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with the scalar finite-element method,” J. Lightwave Technol. 6(6), 1126–1135 (1988).
[CrossRef]

Becker, R. A.

R. A. Becker, R. H. Rediker, and T. A. Lind, “Wide-bandwidth guided-wave electro-optic intensity modulator at λ = 3.39 μm,” Appl. Phys. Lett. 46(9), 809–811 (1985).
[CrossRef]

Beichman, C. A.

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

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(3), 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(13), 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(1), 173–177 (1999).
[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(3), 1259–1264 (2006).
[CrossRef]

G. Li, T. Eckhause, K. A. Winick, J. D. Monnier, and J. P. Berger, “Integrated optic beam combiners in lithium niobate for stellar interferometer,” Proc. SPIE 6268, 626834 (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(3), 1171–1176 (2002).
[CrossRef]

J. P. 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(13), 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(1), 173–177 (1999).
[CrossRef]

Bland-Hawthorn, J.

Booth, A. J.

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, “Detecting nonsolar planets by spinning infrared interferometers,” Nature 274(5673), 780–781 (1978).
[CrossRef]

Breckinridge, J.

J. Breckinridge and C. Lindensmith, “The astronomical search for origins,” Opt. Photonics News 16(2), 24–29 (2005).
[CrossRef]

Broquin, J.-E.

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Brustlein, S.

S. Brustlein, L. Del Rio, A. Tonello, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Laboratory demonstration of an infrared-to-visible up-conversion interferometer for spatial coherence analysis,” Phys. Rev. Lett. 100(15), 153903 (2008).
[CrossRef] [PubMed]

Carleton, N. P.

B. Mennesson, J. M. Mariotti, V. Coudé du 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).

Chiang, K. S.

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3(2), 385–391 (1985).
[CrossRef]

Coudé du Foresto, V.

B. Mennesson, J. M. Mariotti, V. Coudé du 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).

Creech-Eakman, M.

T. ten Brummelaar, M. Creech-Eakman, and J. Monnier, “Probing stars with optical and near-IR interferometry,” Phys. Today 62(6), 28–33 (2009).
[CrossRef]

Danchi, W. C.

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

Del Rio, L.

S. Brustlein, L. Del Rio, A. Tonello, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Laboratory demonstration of an infrared-to-visible up-conversion interferometer for spatial coherence analysis,” Phys. Rev. Lett. 100(15), 153903 (2008).
[CrossRef] [PubMed]

Delage, L.

S. Brustlein, L. Del Rio, A. Tonello, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Laboratory demonstration of an infrared-to-visible up-conversion interferometer for spatial coherence analysis,” Phys. Rev. Lett. 100(15), 153903 (2008).
[CrossRef] [PubMed]

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(3), 1259–1264 (2006).
[CrossRef]

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Diaz, R. T.

Eckhause, T.

G. Li, T. Eckhause, K. A. Winick, J. D. Monnier, and J. P. Berger, “Integrated optic beam combiners in lithium niobate for stellar interferometer,” Proc. SPIE 6268, 626834 (2006).
[CrossRef]

Fridlund, C. V. M.

C. V. M. Fridlund, “The Darwin mission,” Adv. Space Res. 34(3), 613–617 (2004).
[CrossRef]

Gappinger, R. O.

R. O. Gappinger, R. T. Diaz, A. Ksendzov, P. R. Lawson, O. P. Lay, K. M. Liewer, F. M. Loya, S. R. Martin, E. Serabyn, and J. K. Wallace, “Experimental evaluation of achromatic phase shifters for mid-infrared starlight suppression,” Appl. Opt. 48(5), 868–880 (2009).
[CrossRef] [PubMed]

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

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(3), 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(3), 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(3), 1171–1176 (2002).
[CrossRef]

J. P. 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(13), 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(1), 173–177 (1999).
[CrossRef]

Hashimoto, T.

Y. Sakamaki, T. Saida, M. Tamura, T. Hashimoto, and H. Takahashi, “Low loss crosstalk waveguide crossings designed by wavefront matching method,” IEEE Photon. Technol. Lett. 18(19), 2005–2007 (2006).
[CrossRef]

Herrmann, H.

S. Brustlein, L. Del Rio, A. Tonello, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Laboratory demonstration of an infrared-to-visible up-conversion interferometer for spatial coherence analysis,” Phys. Rev. Lett. 100(15), 153903 (2008).
[CrossRef] [PubMed]

Hô, N.

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(3), 1259–1264 (2006).
[CrossRef]

Johnston, K. J.

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
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Kern, P.

J. Bland-Hawthorn and P. Kern, “Astrophotonics: a new era for astronomical instruments,” Opt. Express 17(3), 1880–1884 (2009).
[CrossRef] [PubMed]

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(3), 1259–1264 (2006).
[CrossRef]

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (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(3), 1171–1176 (2002).
[CrossRef]

J. P. 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(13), 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(1), 173–177 (1999).
[CrossRef]

Kirschner, V.

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Kishioka, K.

Korotky, S. K.

W. J. Minford, S. K. Korotky, and R. C. Alferness, “Low-loss Ti:LiNbO3 waveguide bends at λ = 1.3 μm,” IEEE J. Quantum Electron. QE-18, 1802–1806 (1982).
[CrossRef]

Krishnaswami, K.

Ksendzov, A.

R. O. Gappinger, R. T. Diaz, A. Ksendzov, P. R. Lawson, O. P. Lay, K. M. Liewer, F. M. Loya, S. R. Martin, E. Serabyn, and J. K. Wallace, “Experimental evaluation of achromatic phase shifters for mid-infrared starlight suppression,” Appl. Opt. 48(5), 868–880 (2009).
[CrossRef] [PubMed]

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

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L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

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L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
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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(3), 1259–1264 (2006).
[CrossRef]

Lacasse, M. G.

B. Mennesson, J. M. Mariotti, V. Coudé du 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(3), 1171–1176 (2002).
[CrossRef]

Lawson, P. R.

R. O. Gappinger, R. T. Diaz, A. Ksendzov, P. R. Lawson, O. P. Lay, K. M. Liewer, F. M. Loya, S. R. Martin, E. Serabyn, and J. K. Wallace, “Experimental evaluation of achromatic phase shifters for mid-infrared starlight suppression,” Appl. Opt. 48(5), 868–880 (2009).
[CrossRef] [PubMed]

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

Lay, O. P.

R. O. Gappinger, R. T. Diaz, A. Ksendzov, P. R. Lawson, O. P. Lay, K. M. Liewer, F. M. Loya, S. R. Martin, E. Serabyn, and J. K. Wallace, “Experimental evaluation of achromatic phase shifters for mid-infrared starlight suppression,” Appl. Opt. 48(5), 868–880 (2009).
[CrossRef] [PubMed]

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[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(3), 1259–1264 (2006).
[CrossRef]

LeCoarer, E.

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Li, G.

G. Li, T. Eckhause, K. A. Winick, J. D. Monnier, and J. P. Berger, “Integrated optic beam combiners in lithium niobate for stellar interferometer,” Proc. SPIE 6268, 626834 (2006).
[CrossRef]

Liewer, K. M.

Lind, T. A.

R. A. Becker, R. H. Rediker, and T. A. Lind, “Wide-bandwidth guided-wave electro-optic intensity modulator at λ = 3.39 μm,” Appl. Phys. Lett. 46(9), 809–811 (1985).
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J. Breckinridge and C. Lindensmith, “The astronomical search for origins,” Opt. Photonics News 16(2), 24–29 (2005).
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Loya, F. M.

Lunine, J. I.

J. I. Lunine, B. Macintosh, and S. Peale, “The detection and characterization of exoplanets,” Phys. Today 62(5), 46–51 (2009).
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Macintosh, B.

J. I. Lunine, B. Macintosh, and S. Peale, “The detection and characterization of exoplanets,” Phys. Today 62(5), 46–51 (2009).
[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(3), 1259–1264 (2006).
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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(3), 1171–1176 (2002).
[CrossRef]

J. P. 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(13), 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(1), 173–177 (1999).
[CrossRef]

Mariotti, J. M.

B. Mennesson, J. M. Mariotti, V. Coudé du 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).

Martin, S. R.

R. O. Gappinger, R. T. Diaz, A. Ksendzov, P. R. Lawson, O. P. Lay, K. M. Liewer, F. M. Loya, S. R. Martin, E. Serabyn, and J. K. Wallace, “Experimental evaluation of achromatic phase shifters for mid-infrared starlight suppression,” Appl. Opt. 48(5), 868–880 (2009).
[CrossRef] [PubMed]

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

Mazé, G.

B. Mennesson, J. M. Mariotti, V. Coudé du 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. Coudé du 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).

Michelson, A. A.

A. A. Michelson, “Measurement of Jupiter’s satellites by interference,” Nature 45(1155), 160–161 (1891).
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Millan-Gabet, R.

J. P. 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]

Minford, W. J.

W. J. Minford, S. K. Korotky, and R. C. Alferness, “Low-loss Ti:LiNbO3 waveguide bends at λ = 1.3 μm,” IEEE J. Quantum Electron. QE-18, 1802–1806 (1982).
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Monnier, J.

T. ten Brummelaar, M. Creech-Eakman, and J. Monnier, “Probing stars with optical and near-IR interferometry,” Phys. Today 62(6), 28–33 (2009).
[CrossRef]

Monnier, J. D.

G. Li, T. Eckhause, K. A. Winick, J. D. Monnier, and J. P. Berger, “Integrated optic beam combiners in lithium niobate for stellar interferometer,” Proc. SPIE 6268, 626834 (2006).
[CrossRef]

J. D. Monnier, “Asymmetric beam combination for optical interferometry,” Publ. Astron. Soc. Pac. 113(783), 639–645 (2001).
[CrossRef]

Montrosset, I.

E. Strake, G. P. Bava, and I. Montrosset, “Guided modes of Ti:LiNbO3 channel waveguides: a novel quasi-analytical technique in comparison with the scalar finite-element method,” J. Lightwave Technol. 6(6), 1126–1135 (1988).
[CrossRef]

Myers, T. L.

Peale, S.

J. I. Lunine, B. Macintosh, and S. Peale, “The detection and characterization of exoplanets,” Phys. Today 62(5), 46–51 (2009).
[CrossRef]

Perrin, G.

B. Mennesson, J. M. Mariotti, V. Coudé du 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).

Peters, R. D.

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

Phillips, M. C.

Pradel, A.

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Qiao, H.

Rediker, R. H.

R. A. Becker, R. H. Rediker, and T. A. Lind, “Wide-bandwidth guided-wave electro-optic intensity modulator at λ = 3.39 μm,” Appl. Phys. Lett. 46(9), 809–811 (1985).
[CrossRef]

Reynaud, F.

S. Brustlein, L. Del Rio, A. Tonello, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Laboratory demonstration of an infrared-to-visible up-conversion interferometer for spatial coherence analysis,” Phys. Rev. Lett. 100(15), 153903 (2008).
[CrossRef] [PubMed]

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(1), 173–177 (1999).
[CrossRef]

Ridgway, S.

B. Mennesson, J. M. Mariotti, V. Coudé du 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(3), 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(3), 1171–1176 (2002).
[CrossRef]

J. P. 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(13), 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(1), 173–177 (1999).
[CrossRef]

Ruilier, C.

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Saguet, P.

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
[CrossRef]

Saida, T.

Y. Sakamaki, T. Saida, M. Tamura, T. Hashimoto, and H. Takahashi, “Low loss crosstalk waveguide crossings designed by wavefront matching method,” IEEE Photon. Technol. Lett. 18(19), 2005–2007 (2006).
[CrossRef]

Sakamaki, Y.

Y. Sakamaki, T. Saida, M. Tamura, T. Hashimoto, and H. Takahashi, “Low loss crosstalk waveguide crossings designed by wavefront matching method,” IEEE Photon. Technol. Lett. 18(19), 2005–2007 (2006).
[CrossRef]

Schanen, I.

J. P. 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(3), 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(13), 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(1), 173–177 (1999).
[CrossRef]

Scharf, D. P.

P. R. Lawson, O. P. Lay, S. R. Martin, R. D. Peters, R. O. Gappinger, A. Ksendzov, D. P. Scharf, A. J. Booth, C. A. Beichman, E. Serabyn, K. J. Johnston, and W. C. Danchi, “Terrestrial planet finder interferometer 2007-2008 progress and plans,” Proc. SPIE 7013, 70132N (2008).
[CrossRef]

Schmidt, R. V.

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(3), 1259–1264 (2006).
[CrossRef]

Serabyn, E.

R. O. Gappinger, R. T. Diaz, A. Ksendzov, P. R. Lawson, O. P. Lay, K. M. Liewer, F. M. Loya, S. R. Martin, E. Serabyn, and J. K. Wallace, “Experimental evaluation of achromatic phase shifters for mid-infrared starlight suppression,” Appl. Opt. 48(5), 868–880 (2009).
[CrossRef] [PubMed]

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Y. Sakamaki, T. Saida, M. Tamura, T. Hashimoto, and H. Takahashi, “Low loss crosstalk waveguide crossings designed by wavefront matching method,” IEEE Photon. Technol. Lett. 18(19), 2005–2007 (2006).
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S. Brustlein, L. Del Rio, A. Tonello, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Laboratory demonstration of an infrared-to-visible up-conversion interferometer for spatial coherence analysis,” Phys. Rev. Lett. 100(15), 153903 (2008).
[CrossRef] [PubMed]

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J. P. 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).
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[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(3), 1171–1176 (2002).
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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(3), 1259–1264 (2006).
[CrossRef]

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Astron. Astrophys. Suppl. Ser. (1)

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(1), 173–177 (1999).
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IEEE Photon. Technol. Lett. (1)

Y. Sakamaki, T. Saida, M. Tamura, T. Hashimoto, and H. Takahashi, “Low loss crosstalk waveguide crossings designed by wavefront matching method,” IEEE Photon. Technol. Lett. 18(19), 2005–2007 (2006).
[CrossRef]

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[CrossRef] [PubMed]

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T. ten Brummelaar, M. Creech-Eakman, and J. Monnier, “Probing stars with optical and near-IR interferometry,” Phys. Today 62(6), 28–33 (2009).
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[CrossRef]

G. Li, T. Eckhause, K. A. Winick, J. D. Monnier, and J. P. Berger, “Integrated optic beam combiners in lithium niobate for stellar interferometer,” Proc. SPIE 6268, 626834 (2006).
[CrossRef]

L. Labadie, L. Abel-Tiberini, E. LeCoarer, C. Vigreuz-Bercovici, B. Arezki, M. Barillot, J.-E. Broquin, A. Delboulbé, P. Kern, V. Kirschner, P. Labeye, A. Pradel, C. Ruilier, and P. Saguet, “Recent progress in mid infrared integrated optics for nulling interferometry,” Proc. SPIE 6268, 62682E (2006).
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Figures (9)

Fig. 1
Fig. 1

Layout of (a) fabricated three-beam combiner (b) two-beam combiner for device characterization (figures not drawn to scale).

Fig. 2
Fig. 2

(a) Comparison of the measured effective indices and calculated effective indices for a Gaussian profile (b) Extraordinary index profile of planar waveguide.

Fig. 3
Fig. 3

(a) Contour map of Refractive index (b) Computed TE-mode profile.

Fig. 9
Fig. 9

White-light fringe and theoretical fringe with 500 nm bandwidth.

Fig. 4
Fig. 4

Two semi-circular arcs with radius of curvature R.

Fig. 5
Fig. 5

Bending loss measurements and exponential fit.

Fig. 6
Fig. 6

Experimentally determined coupling characteristics.

Fig. 7
Fig. 7

The oscilloscope traces of applied MZI voltage and output signal power.

Fig. 8
Fig. 8

Experimental set-up for measurement of white-light interferograms.

Equations (7)

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n(x,z)=ns+Δn12[erf(W/2+zDz)+erf(W/2zDz)]exp(x2Dx2)
α(R)=C1exp(C2R) (dB/cm)
η=sin2(κLd+ϕ)
|Aout|2=|Ain|2[1C(λ)]+|Bin|2C(λ)2|Ain||Bin|C(λ)[1C(λ)]sin(AinBin+φint(λ)+θext(λ))|Bout|2=|Ain|2[1C(λ)]+|Bin|2C(λ)+2|Ain||Bin|C(λ)[1C(λ)]sin(AinBin+φint(λ)+θext(λ))
θext(λ)=2πλΔneo(λ)Le
Δϕ=2Γeffπn3rVDgapLeλ
Io=1Δλλ0Δλ2λ0+Δλ212cos(2πΔnLeλ)dλ12cosΔϕ0sin(Δϕ0Δλ2λ0)Δϕ0Δλ2λ0

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