Abstract

Nulling interferometry is an astronomical technique that combines equal wavefronts to achieve a deep rejection ratio of an on-axis star, and that could permit to detect Earth-like planets in the mid-infrared band [5 – 20 μm]. Similarly to what is done in the near-infrared, high frequencies spatial filtering of the incoming beams can be achieved using single-mode waveguides operating in the mid-infrared. An appreciable reduction of the instrumental complexity is also possible using integrated optics (IO) devices in this spectral range. The relative lack of single-mode guided optics in the mid-infrared has motivated the present technological study to demonstrate the feasibility of dielectric waveguides functioning at longer wavelengths. We propose to use selenide and telluride components to pursue the development of more complex IO functions.

© 2006 Optical Society of America

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References

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  1. V. Coudé du Foresto, G. Perrin, J.-M. Mariotti, M. Lacasse, and W. Traub, "The FLUOR/IOTA fiber stellar interferometer," in Proceedings of Integrated Optics for Astronomical Interferometry, P. Kern and F. Malbet, eds. (Grenoble, France, 1997), pp. 115-125.
  2. J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
    [CrossRef]
  3. F. Malbet, J.-P. Berger, P. Garcia, P. Kern, K. Perraut, M. Benisty, L. Jocou, E. Herwats, J,-B. Lebouquin, P. Labeye, E. Le Coarer, and O. Preis, "VITRUV - Imaging close environments of stars and galaxies with the VLTI at milli-arcsec resolution," presented at the VLTI Workshop on Optical and IR Interferometry, Garching bei Munchen, Germany, 4-8 April 2005.
  4. G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
    [CrossRef] [PubMed]
  5. C. V. M. Fridlund, "The Darwin mission," Adv. Space Res. 34,613-617 (2004).
    [CrossRef]
  6. R. N. Bracewell, "Detecting non-solar planets by spinning infrared interferometer," Nature (London) 274,780-781 (1978).
    [CrossRef]
  7. J. R. P. Angel, A. Y. S. Cheng, and N. J. Woolf, "A space telescope for infrared spectroscopy of earth-like planets," Nature (London) 322,341-343 (1986).
    [CrossRef]
  8. B. Mennesson, M. Ollivier, and C. Ruilier, "Use of single-mode waveguides to correct the optical defects of a nulling interferometer," J. Opt. Soc. Am. A 19,596-602 (2002).
    [CrossRef]
  9. P. Bordé, G. Perrin, A. Amy-Klein, C. Daussy, and G. Mazé, "Updated results on prototype chalcogenide fibers for 10 μm wavefront spatial filtering," in Proceedings of DARWIN/TPF and the Search for Extra-solar Terrestrial Planets, M. Fridlund and T. Henning, eds. (Noordwijk, Netherlands, 2003), pp. 371-374.
  10. U. J. Wehmeier, M. R. Swain, C. Y. D. D’Aubigny, D. R. Golish and C. K. Walker, "The potential of conductive waveguides for nulling interferometry," in New Frontiers for Stellar Interferometry, W. A. Traub, ed., Proc. SPIE 5491, 1435-1443 (2004).
    [CrossRef]
  11. O. Wallner, V. G. Artjuschenko, and R. Flatscher, "Development of silver-halide single-mode fibers for modal filtering in the mid-infrared," in New Frontiers for Stellar Interferometry, W. A. Traub, ed., Proc. SPIE 5491, 636-646 (2004).
    [CrossRef]
  12. L. Labadie, P. Labeye, P. Kern, I. Schanen-Duport, B. Arezki, and J.-E. Broquin, "Modal filtering for nulling interferometry - First single-mode conductive waveguides in the mid-infrared," Astron. &Astrophys. 460,1265-1266 (2006).
    [CrossRef]
  13. D. L. Lee, Electromagnetic Principles of Integrated Optics (John Wiley and Sons Ltd., 1986).
  14. V. Kokorina, Glasses for Infrared Optics (CRC Press, 1996).
  15. P. K. Tien, R. Ulrich, and R. J. Martin, "Modes of propagating light waves in thin deposited semiconductor films," Appl. Phys. Lett. 14,291-294 (1969).
    [CrossRef]
  16. P. K. Tien, "Light waves in thin films and integrated optics," Appl. Opt. 10,2395-2413 (1971).
    [CrossRef] [PubMed]
  17. H. J. Lee, C. H. Henry, K. J. Orlowsky, R. F. Kazarinov, and T. Y. Kometani, "Refractive-index dispersion of phosphosilicate glass, thermal oxide and silicon nitride films on silicon," Appl. Opt. 27,4104-4109 (1988).
    [CrossRef] [PubMed]
  18. J. M. White, and P. F. Heidrich, "Optical waveguide refractive index profiles determined from measurements of mode indices: a simple analysis," Appl. Opt. 15,151-155 (1976).
    [CrossRef] [PubMed]
  19. E. D. Palik, Handbook of Optical Constants (Academic Press New York, 1985).

2006 (2)

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

L. Labadie, P. Labeye, P. Kern, I. Schanen-Duport, B. Arezki, and J.-E. Broquin, "Modal filtering for nulling interferometry - First single-mode conductive waveguides in the mid-infrared," Astron. &Astrophys. 460,1265-1266 (2006).
[CrossRef]

2004 (1)

C. V. M. Fridlund, "The Darwin mission," Adv. Space Res. 34,613-617 (2004).
[CrossRef]

2002 (1)

2001 (1)

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

1988 (1)

1986 (1)

J. R. P. Angel, A. Y. S. Cheng, and N. J. Woolf, "A space telescope for infrared spectroscopy of earth-like planets," Nature (London) 322,341-343 (1986).
[CrossRef]

1978 (1)

R. N. Bracewell, "Detecting non-solar planets by spinning infrared interferometer," Nature (London) 274,780-781 (1978).
[CrossRef]

1976 (1)

1971 (1)

1969 (1)

P. K. Tien, R. Ulrich, and R. J. Martin, "Modes of propagating light waves in thin deposited semiconductor films," Appl. Phys. Lett. 14,291-294 (1969).
[CrossRef]

Angel, J. R. P.

J. R. P. Angel, A. Y. S. Cheng, and N. J. Woolf, "A space telescope for infrared spectroscopy of earth-like planets," Nature (London) 322,341-343 (1986).
[CrossRef]

Arezki, B.

L. Labadie, P. Labeye, P. Kern, I. Schanen-Duport, B. Arezki, and J.-E. Broquin, "Modal filtering for nulling interferometry - First single-mode conductive waveguides in the mid-infrared," Astron. &Astrophys. 460,1265-1266 (2006).
[CrossRef]

Berger, J.-P.

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, "Detecting non-solar planets by spinning infrared interferometer," Nature (London) 274,780-781 (1978).
[CrossRef]

Broquin, J.-E.

L. Labadie, P. Labeye, P. Kern, I. Schanen-Duport, B. Arezki, and J.-E. Broquin, "Modal filtering for nulling interferometry - First single-mode conductive waveguides in the mid-infrared," Astron. &Astrophys. 460,1265-1266 (2006).
[CrossRef]

Chaffee, F.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Cheng, A. Y. S.

J. R. P. Angel, A. Y. S. Cheng, and N. J. Woolf, "A space telescope for infrared spectroscopy of earth-like planets," Nature (London) 322,341-343 (1986).
[CrossRef]

Fridlund, C. V. M.

C. V. M. Fridlund, "The Darwin mission," Adv. Space Res. 34,613-617 (2004).
[CrossRef]

Gathright, J.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Gu´erin, J.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Haguenauer, P.

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Heidrich, P. F.

Henry, C. H.

Hrynevych, M.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Kazarinov, R. F.

Kern, P.

L. Labadie, P. Labeye, P. Kern, I. Schanen-Duport, B. Arezki, and J.-E. Broquin, "Modal filtering for nulling interferometry - First single-mode conductive waveguides in the mid-infrared," Astron. &Astrophys. 460,1265-1266 (2006).
[CrossRef]

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Kometani, T. Y.

Kotani, T.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

L´ena, P.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Labadie, L.

L. Labadie, P. Labeye, P. Kern, I. Schanen-Duport, B. Arezki, and J.-E. Broquin, "Modal filtering for nulling interferometry - First single-mode conductive waveguides in the mid-infrared," Astron. &Astrophys. 460,1265-1266 (2006).
[CrossRef]

Labeye, P.

L. Labadie, P. Labeye, P. Kern, I. Schanen-Duport, B. Arezki, and J.-E. Broquin, "Modal filtering for nulling interferometry - First single-mode conductive waveguides in the mid-infrared," Astron. &Astrophys. 460,1265-1266 (2006).
[CrossRef]

Lai, O.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Lee, H. J.

LeMignant, D.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Malbet, F.

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Martin, R. J.

P. K. Tien, R. Ulrich, and R. J. Martin, "Modes of propagating light waves in thin deposited semiconductor films," Appl. Phys. Lett. 14,291-294 (1969).
[CrossRef]

Mennesson, B.

Millan-Gabet, R.

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Ollivier, M.

Orlowsky, K. J.

Perraut, K.

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Perrin, G.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Ruilier, C.

Schanen, I.

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Schanen-Duport, I.

L. Labadie, P. Labeye, P. Kern, I. Schanen-Duport, B. Arezki, and J.-E. Broquin, "Modal filtering for nulling interferometry - First single-mode conductive waveguides in the mid-infrared," Astron. &Astrophys. 460,1265-1266 (2006).
[CrossRef]

Severi, M.

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Tien, P. K.

P. K. Tien, "Light waves in thin films and integrated optics," Appl. Opt. 10,2395-2413 (1971).
[CrossRef] [PubMed]

P. K. Tien, R. Ulrich, and R. J. Martin, "Modes of propagating light waves in thin deposited semiconductor films," Appl. Phys. Lett. 14,291-294 (1969).
[CrossRef]

Traub, W.

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Ulrich, R.

P. K. Tien, R. Ulrich, and R. J. Martin, "Modes of propagating light waves in thin deposited semiconductor films," Appl. Phys. Lett. 14,291-294 (1969).
[CrossRef]

Vergnole, S.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

White, J. M.

Wizinowich, P. L.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Woillez, J.

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Woolf, N. J.

J. R. P. Angel, A. Y. S. Cheng, and N. J. Woolf, "A space telescope for infrared spectroscopy of earth-like planets," Nature (London) 322,341-343 (1986).
[CrossRef]

Adv. Space Res. (1)

C. V. M. Fridlund, "The Darwin mission," Adv. Space Res. 34,613-617 (2004).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

P. K. Tien, R. Ulrich, and R. J. Martin, "Modes of propagating light waves in thin deposited semiconductor films," Appl. Phys. Lett. 14,291-294 (1969).
[CrossRef]

Astron. & Astrophys. (1)

J.-P. Berger, P. Haguenauer, P. Kern, K. 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.Supp. 376,L31-L34 (2001).
[CrossRef]

Astron. &Astrophys. (1)

L. Labadie, P. Labeye, P. Kern, I. Schanen-Duport, B. Arezki, and J.-E. Broquin, "Modal filtering for nulling interferometry - First single-mode conductive waveguides in the mid-infrared," Astron. &Astrophys. 460,1265-1266 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nature (London) (2)

R. N. Bracewell, "Detecting non-solar planets by spinning infrared interferometer," Nature (London) 274,780-781 (1978).
[CrossRef]

J. R. P. Angel, A. Y. S. Cheng, and N. J. Woolf, "A space telescope for infrared spectroscopy of earth-like planets," Nature (London) 322,341-343 (1986).
[CrossRef]

Science (1)

G. Perrin, J. Woillez, O. Lai, J. Gu´erin, T. Kotani, P. L. Wizinowich, D. LeMignant,M. Hrynevych, J. Gathright, P. L´ena, F. Chaffee, and S. Vergnole, "Interferometric coupling of the Keck telescopes with single-mode fibers," Science 311,194 (2006).
[CrossRef] [PubMed]

Other (8)

V. Coudé du Foresto, G. Perrin, J.-M. Mariotti, M. Lacasse, and W. Traub, "The FLUOR/IOTA fiber stellar interferometer," in Proceedings of Integrated Optics for Astronomical Interferometry, P. Kern and F. Malbet, eds. (Grenoble, France, 1997), pp. 115-125.

F. Malbet, J.-P. Berger, P. Garcia, P. Kern, K. Perraut, M. Benisty, L. Jocou, E. Herwats, J,-B. Lebouquin, P. Labeye, E. Le Coarer, and O. Preis, "VITRUV - Imaging close environments of stars and galaxies with the VLTI at milli-arcsec resolution," presented at the VLTI Workshop on Optical and IR Interferometry, Garching bei Munchen, Germany, 4-8 April 2005.

P. Bordé, G. Perrin, A. Amy-Klein, C. Daussy, and G. Mazé, "Updated results on prototype chalcogenide fibers for 10 μm wavefront spatial filtering," in Proceedings of DARWIN/TPF and the Search for Extra-solar Terrestrial Planets, M. Fridlund and T. Henning, eds. (Noordwijk, Netherlands, 2003), pp. 371-374.

U. J. Wehmeier, M. R. Swain, C. Y. D. D’Aubigny, D. R. Golish and C. K. Walker, "The potential of conductive waveguides for nulling interferometry," in New Frontiers for Stellar Interferometry, W. A. Traub, ed., Proc. SPIE 5491, 1435-1443 (2004).
[CrossRef]

O. Wallner, V. G. Artjuschenko, and R. Flatscher, "Development of silver-halide single-mode fibers for modal filtering in the mid-infrared," in New Frontiers for Stellar Interferometry, W. A. Traub, ed., Proc. SPIE 5491, 636-646 (2004).
[CrossRef]

D. L. Lee, Electromagnetic Principles of Integrated Optics (John Wiley and Sons Ltd., 1986).

V. Kokorina, Glasses for Infrared Optics (CRC Press, 1996).

E. D. Palik, Handbook of Optical Constants (Academic Press New York, 1985).

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Figures (3)

Fig. 1.
Fig. 1.

Typical prism coupler setup. When we are in the condition of total reflection at the base of the prism (θ p smaller than critical angle θ c), the modes of the slab waveguide can be excited through the evanescent field existing in the air gap. The air gap thickness is controlled with a pushing screw at the back of the waveguide. A and B are the angles of the prism. θ 1,m and θ 2,m are the algebraic angles with respect to the normal of input and output prism facets corresponding to the excitation of mode m. θ m is the angle of the propagating ray corresponding to mode m.

Fig. 2.
Fig. 2.

Experimental arrangement of the mid infrared m-lines experiment: QW, quarter-wave plate; M, mirror; BS, beam-splitter; A, absorbing refractory brick; P, polarizer; B, beam-expander; R, reticle; F, focusing optics; S, system prism – film; C, infrared camera.

Fig. 3.
Fig. 3.

Output of the m-lines experiment in the mid-IR (left) and in the near-IR (right). The image of the reticule is used to point the black lines by turning the prism stage. The mid-IR pattern presents a circular diffraction pattern due to the high coherence of the laser source.

Tables (4)

Tables Icon

Table 1. TE mode indices obtained experimentally on sample As-1 at λ=1.196 μm with the m-lines setup. The error on the measured angle is ±0.01°. The error on experimental neff is ±1.3×l0-4. We found a maximum coefficient ρ=-0.99923 for the presented mode numbering. The silicon prism has np=3.5193 and B=44.8416°. The substrate refractive index at λ=l. 196 μm is 2.47[19]. Note that the two first modes could not be experimentally observed with the silicon prism and are labelled as ”a” for ”absent”. The sign ”N/A” means ”not applicable”.

Tables Icon

Table 2. TE mode indices obtained experimentally on samples As-1 and As-2 at λ=10.6 μm. The error on measured angle is ±0.01°. The accuracies on prism index and angle lead to an error on experimental neff of ±10-3. The maximum coefficient is ρ=-0.99982 for the given mode numbering. The prism has np=4.00±0.001 and B=30.0±0.01°. The substrate refractive index at λ=10.6 μm is 2.38. The sign ”N/A” means ”not applicable”.

Tables Icon

Table 3. TM (up) and TE (down) mode indices obtained experimentally on sample TAS-1 at λ=10.6 μm. The error on measured angle is ±0.01°. The accuracies on prism index and angle lead to an error on experimental neff of ±10-3. The same germanium prism as in Table 2 is used. The sign ”a” means ”absent” and ”N/A” means ”not applicable”.

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Table 4. TE mode indices obtained on sample TAS-2 at λ=10.6 μm. The error on measured angle is ±0.01° and the accuracies on prism index and angle lead to an error on experimental neff of ±10-3.

Equations (4)

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d > d m = m π + arctan ( g n sub 2 n sup 2 n c 2 n sub 2 ) k n c 2 n sub 2
n p cos ( θ p ) = n c cos ( θ m )
n eff , m = n p sin ( arcsin ( sin ( θ 2 , m ) n p ) + B )
k d n c 2 n eff , m 2 arctan ( g n eff , m 2 n sub 2 n c 2 n eff , m 2 ) arctan ( g n eff , m 2 n sup 2 n c 2 n eff , m 2 ) = m π

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