Abstract

High-contrast imaging systems with a stellar halo suppression level of 10−10 are required for direct detection of Earth-like extra-solar planets. We investigated a novel high-contrast imaging system with an unbalanced nulling interferometer (UNI) followed by phase and amplitude correction (PAC), which not only can reduce starlight but also can suppress the speckle level caused by wavefront aberrations. We successfully demonstrated that wavefront aberrations were sufficiently magnified by the UNI and the magnified aberrations were effectively corrected in amplitude and phase with two deformable mirrors. We confirmed that the suppression level of the speckle pattern with the proposed optics was beyond the limit of the adaptive optics performance.

© 2011 OSA

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  1. R. N. Bracewell, “Detecting nonsolar planets by spinning infrared interferometer,” Nature 274(5673), 780–781 (1978).
    [CrossRef]
  2. P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interferocoronagraphy, I. Theoretical capabilities for ground-based observations,” Astron. Astrophys. Suppl. Ser. 141, 319–329 (2000).
    [CrossRef]
  3. K. Yokochi, A. V. Tavrov, J. Nishikawa, N. Murakami, L. Abe, M. Tamura, M. Takeda, and T. Kurokawa, “Achromatic deep nulling with a three-dimensional Sagnac interferometer,” Opt. Lett. 34(13), 1985–1987 (2009).
    [CrossRef] [PubMed]
  4. M. Shao, J. K. Wallace, B. M. Levine, and D. Liu, “Visible nulling coronagraph,” Proc. SPIE 5487, 1296–1303 (2004).
    [CrossRef]
  5. N. Murakami, K. Yokochi, J. Nishikawa, M. Tamura, T. Kurokawa, M. Takeda, and N. Baba, “Polarization interferometric nulling coronagraph for high-contrast imaging,” Appl. Opt. 49(16), D106–D114 (2010).
    [CrossRef] [PubMed]
  6. O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. Suppl. Ser. 167, 81–99 (2006).
    [CrossRef]
  7. P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
    [CrossRef] [PubMed]
  8. C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
    [CrossRef] [PubMed]
  9. A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
    [CrossRef]
  10. C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
    [CrossRef]
  11. M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570(2), 900–908 (2002).
    [CrossRef]
  12. F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
    [CrossRef]
  13. A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, “On representing and correcting wavefront errors in high-contrast imaging systems,” J. Opt. Soc. Am. A 23(5), 1063–1073 (2006).
    [CrossRef]
  14. R. Galicher, P. Baudoz, G. Rousset, J. Totems, and M. Mas, “Self-coherent camera as a focal plane wavefront sensor: simulations,” Astron. Astrophys. 509, A31 (2010).
    [CrossRef]
  15. J. Nishikawa, L. Abe, N. Murakami, and T. Kotani, “Precise wavefront correction with an unbalanced nulling interferometer for exo-planet imaging coronagraphs,” Astron. Astrophys. 489(3), 1389–1398 (2008).
    [CrossRef]
  16. N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase-mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
    [CrossRef]

2010

N. Murakami, K. Yokochi, J. Nishikawa, M. Tamura, T. Kurokawa, M. Takeda, and N. Baba, “Polarization interferometric nulling coronagraph for high-contrast imaging,” Appl. Opt. 49(16), D106–D114 (2010).
[CrossRef] [PubMed]

R. Galicher, P. Baudoz, G. Rousset, J. Totems, and M. Mas, “Self-coherent camera as a focal plane wavefront sensor: simulations,” Astron. Astrophys. 509, A31 (2010).
[CrossRef]

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase-mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

2009

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

K. Yokochi, A. V. Tavrov, J. Nishikawa, N. Murakami, L. Abe, M. Tamura, M. Takeda, and T. Kurokawa, “Achromatic deep nulling with a three-dimensional Sagnac interferometer,” Opt. Lett. 34(13), 1985–1987 (2009).
[CrossRef] [PubMed]

2008

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

J. Nishikawa, L. Abe, N. Murakami, and T. Kotani, “Precise wavefront correction with an unbalanced nulling interferometer for exo-planet imaging coronagraphs,” Astron. Astrophys. 489(3), 1389–1398 (2008).
[CrossRef]

2006

A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, “On representing and correcting wavefront errors in high-contrast imaging systems,” J. Opt. Soc. Am. A 23(5), 1063–1073 (2006).
[CrossRef]

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. Suppl. Ser. 167, 81–99 (2006).
[CrossRef]

2004

M. Shao, J. K. Wallace, B. M. Levine, and D. Liu, “Visible nulling coronagraph,” Proc. SPIE 5487, 1296–1303 (2004).
[CrossRef]

2002

M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570(2), 900–908 (2002).
[CrossRef]

2000

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interferocoronagraphy, I. Theoretical capabilities for ground-based observations,” Astron. Astrophys. Suppl. Ser. 141, 319–329 (2000).
[CrossRef]

1995

F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
[CrossRef]

1978

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

Abe, L.

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase-mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

K. Yokochi, A. V. Tavrov, J. Nishikawa, N. Murakami, L. Abe, M. Tamura, M. Takeda, and T. Kurokawa, “Achromatic deep nulling with a three-dimensional Sagnac interferometer,” Opt. Lett. 34(13), 1985–1987 (2009).
[CrossRef] [PubMed]

J. Nishikawa, L. Abe, N. Murakami, and T. Kotani, “Precise wavefront correction with an unbalanced nulling interferometer for exo-planet imaging coronagraphs,” Astron. Astrophys. 489(3), 1389–1398 (2008).
[CrossRef]

Allard, F.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Avitzour, Y.

Baba, N.

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase-mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

N. Murakami, K. Yokochi, J. Nishikawa, M. Tamura, T. Kurokawa, M. Takeda, and N. Baba, “Polarization interferometric nulling coronagraph for high-contrast imaging,” Appl. Opt. 49(16), D106–D114 (2010).
[CrossRef] [PubMed]

Barman, T.

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

Baudoz, P.

R. Galicher, P. Baudoz, G. Rousset, J. Totems, and M. Mas, “Self-coherent camera as a focal plane wavefront sensor: simulations,” Astron. Astrophys. 509, A31 (2010).
[CrossRef]

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interferocoronagraphy, I. Theoretical capabilities for ground-based observations,” Astron. Astrophys. Suppl. Ser. 141, 319–329 (2000).
[CrossRef]

Bracewell, R. N.

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

Carson, J.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Charton, J.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Chauvin, G.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Chiang, E.

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

Clampin, M.

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

Collins, B.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. Suppl. Ser. 167, 81–99 (2006).
[CrossRef]

Doyon, R.

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

Egner, S.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Ehrenreich, D.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Feldt, M.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Fitzgerald, M. P.

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

Fusco, T.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Galicher, R.

R. Galicher, P. Baudoz, G. Rousset, J. Totems, and M. Mas, “Self-coherent camera as a focal plane wavefront sensor: simulations,” Astron. Astrophys. 509, A31 (2010).
[CrossRef]

Gay, J.

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interferocoronagraphy, I. Theoretical capabilities for ground-based observations,” Astron. Astrophys. Suppl. Ser. 141, 319–329 (2000).
[CrossRef]

Gendron, E.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Give’on, A.

Goto, M.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Graham, J. R.

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

Gratadour, D.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Guyon, O.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. Suppl. Ser. 167, 81–99 (2006).
[CrossRef]

Hashimoto, J.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Hayano, Y.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Henning, T.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Hodapp, K. W.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Janson, M.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Kalas, P.

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

Kandori, R.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Kasdin, N. J.

Kite, E. S.

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

Klahr, H.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Kotani, T.

J. Nishikawa, L. Abe, N. Murakami, and T. Kotani, “Precise wavefront correction with an unbalanced nulling interferometer for exo-planet imaging coronagraphs,” Astron. Astrophys. 489(3), 1389–1398 (2008).
[CrossRef]

Krist, J.

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

Kuchner, M. J.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. Suppl. Ser. 167, 81–99 (2006).
[CrossRef]

M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570(2), 900–908 (2002).
[CrossRef]

Kudo, T.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Kurokawa, T.

Kusakabe, N.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Lacombe, F.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Lafrenière, D.

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

Lagrange, A.-M.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Levine, B. M.

M. Shao, J. K. Wallace, B. M. Levine, and D. Liu, “Visible nulling coronagraph,” Proc. SPIE 5487, 1296–1303 (2004).
[CrossRef]

Liu, D.

M. Shao, J. K. Wallace, B. M. Levine, and D. Liu, “Visible nulling coronagraph,” Proc. SPIE 5487, 1296–1303 (2004).
[CrossRef]

Macintosh, B.

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

Malbet, F.

F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
[CrossRef]

Marois, C.

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

Mas, M.

R. Galicher, P. Baudoz, G. Rousset, J. Totems, and M. Mas, “Self-coherent camera as a focal plane wavefront sensor: simulations,” Astron. Astrophys. 509, A31 (2010).
[CrossRef]

McElwain, M.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Montri, J.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Mordasini, C.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Morino, J.-I.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Mouillet, D.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Mugnier, L.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Murakami, N.

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase-mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

N. Murakami, K. Yokochi, J. Nishikawa, M. Tamura, T. Kurokawa, M. Takeda, and N. Baba, “Polarization interferometric nulling coronagraph for high-contrast imaging,” Appl. Opt. 49(16), D106–D114 (2010).
[CrossRef] [PubMed]

K. Yokochi, A. V. Tavrov, J. Nishikawa, N. Murakami, L. Abe, M. Tamura, M. Takeda, and T. Kurokawa, “Achromatic deep nulling with a three-dimensional Sagnac interferometer,” Opt. Lett. 34(13), 1985–1987 (2009).
[CrossRef] [PubMed]

J. Nishikawa, L. Abe, N. Murakami, and T. Kotani, “Precise wavefront correction with an unbalanced nulling interferometer for exo-planet imaging coronagraphs,” Astron. Astrophys. 489(3), 1389–1398 (2008).
[CrossRef]

Nishikawa, J.

N. Murakami, K. Yokochi, J. Nishikawa, M. Tamura, T. Kurokawa, M. Takeda, and N. Baba, “Polarization interferometric nulling coronagraph for high-contrast imaging,” Appl. Opt. 49(16), D106–D114 (2010).
[CrossRef] [PubMed]

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase-mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

K. Yokochi, A. V. Tavrov, J. Nishikawa, N. Murakami, L. Abe, M. Tamura, M. Takeda, and T. Kurokawa, “Achromatic deep nulling with a three-dimensional Sagnac interferometer,” Opt. Lett. 34(13), 1985–1987 (2009).
[CrossRef] [PubMed]

J. Nishikawa, L. Abe, N. Murakami, and T. Kotani, “Precise wavefront correction with an unbalanced nulling interferometer for exo-planet imaging coronagraphs,” Astron. Astrophys. 489(3), 1389–1398 (2008).
[CrossRef]

Patience, J.

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

Pluzhnik, E. A.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. Suppl. Ser. 167, 81–99 (2006).
[CrossRef]

Rabbia, Y.

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interferocoronagraphy, I. Theoretical capabilities for ground-based observations,” Astron. Astrophys. Suppl. Ser. 141, 319–329 (2000).
[CrossRef]

Rabou, P.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Ridgway, S. T.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. Suppl. Ser. 167, 81–99 (2006).
[CrossRef]

Rouan, D.

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Rousset, G.

R. Galicher, P. Baudoz, G. Rousset, J. Totems, and M. Mas, “Self-coherent camera as a focal plane wavefront sensor: simulations,” Astron. Astrophys. 509, A31 (2010).
[CrossRef]

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Shao, M.

M. Shao, J. K. Wallace, B. M. Levine, and D. Liu, “Visible nulling coronagraph,” Proc. SPIE 5487, 1296–1303 (2004).
[CrossRef]

F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
[CrossRef]

Song, I.

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

Stapelfeldt, K.

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

Suto, H.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Suzuki, R.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Takeda, M.

Tamura, M.

N. Murakami, K. Yokochi, J. Nishikawa, M. Tamura, T. Kurokawa, M. Takeda, and N. Baba, “Polarization interferometric nulling coronagraph for high-contrast imaging,” Appl. Opt. 49(16), D106–D114 (2010).
[CrossRef] [PubMed]

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase-mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

K. Yokochi, A. V. Tavrov, J. Nishikawa, N. Murakami, L. Abe, M. Tamura, M. Takeda, and T. Kurokawa, “Achromatic deep nulling with a three-dimensional Sagnac interferometer,” Opt. Lett. 34(13), 1985–1987 (2009).
[CrossRef] [PubMed]

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Tavrov, A. V.

Thalmann, C.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

Totems, J.

R. Galicher, P. Baudoz, G. Rousset, J. Totems, and M. Mas, “Self-coherent camera as a focal plane wavefront sensor: simulations,” Astron. Astrophys. 509, A31 (2010).
[CrossRef]

Traub, W. A.

M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570(2), 900–908 (2002).
[CrossRef]

Vanderbei, R. J.

Wallace, J. K.

M. Shao, J. K. Wallace, B. M. Levine, and D. Liu, “Visible nulling coronagraph,” Proc. SPIE 5487, 1296–1303 (2004).
[CrossRef]

Yokochi, K.

Yu, J. W.

F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
[CrossRef]

Zuckerman, B.

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

Appl. Opt.

Astron. Astrophys.

R. Galicher, P. Baudoz, G. Rousset, J. Totems, and M. Mas, “Self-coherent camera as a focal plane wavefront sensor: simulations,” Astron. Astrophys. 509, A31 (2010).
[CrossRef]

J. Nishikawa, L. Abe, N. Murakami, and T. Kotani, “Precise wavefront correction with an unbalanced nulling interferometer for exo-planet imaging coronagraphs,” Astron. Astrophys. 489(3), 1389–1398 (2008).
[CrossRef]

A.-M. Lagrange, D. Gratadour, G. Chauvin, T. Fusco, D. Ehrenreich, D. Mouillet, G. Rousset, D. Rouan, F. Allard, E. Gendron, J. Charton, L. Mugnier, P. Rabou, J. Montri, and F. Lacombe, “A probable giant planet imaged in the β Pictoris disk,” Astron. Astrophys. 493(2), L21–L25 (2009).
[CrossRef]

Astron. Astrophys. Suppl. Ser.

P. Baudoz, Y. Rabbia, and J. Gay, “Achromatic interferocoronagraphy, I. Theoretical capabilities for ground-based observations,” Astron. Astrophys. Suppl. Ser. 141, 319–329 (2000).
[CrossRef]

Astrophys. J.

C. Thalmann, J. Carson, M. Janson, M. Goto, M. McElwain, S. Egner, M. Feldt, J. Hashimoto, Y. Hayano, T. Henning, K. W. Hodapp, R. Kandori, H. Klahr, T. Kudo, N. Kusakabe, C. Mordasini, J.-I. Morino, H. Suto, R. Suzuki, and M. Tamura, “Discovery of the coldest imaged companion of a sun-like star,” Astrophys. J. 707(2), L123–L127 (2009).
[CrossRef]

M. J. Kuchner and W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570(2), 900–908 (2002).
[CrossRef]

N. Murakami, J. Nishikawa, K. Yokochi, M. Tamura, N. Baba, and L. Abe, “Achromatic eight-octant phase-mask coronagraph using photonic crystal,” Astrophys. J. 714(1), 772–777 (2010).
[CrossRef]

Astrophys. J. Suppl. Ser.

O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. Suppl. Ser. 167, 81–99 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Nature

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

Opt. Lett.

Proc. SPIE

M. Shao, J. K. Wallace, B. M. Levine, and D. Liu, “Visible nulling coronagraph,” Proc. SPIE 5487, 1296–1303 (2004).
[CrossRef]

Publ. Astron. Soc. Pac.

F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
[CrossRef]

Science

P. Kalas, J. R. Graham, E. Chiang, M. P. Fitzgerald, M. Clampin, E. S. Kite, K. Stapelfeldt, C. Marois, and J. Krist, “Optical images of an exosolar planet 25 light-years from Earth,” Science 322(5906), 1345–1348 (2008).
[CrossRef] [PubMed]

C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafrenière, and R. Doyon, “Direct imaging of multiple planets orbiting the star HR 8799,” Science 322(5906), 1348–1352 (2008).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Schematic optical layout of a high-contrast imaging system consisting of a first AO, an unbalanced nulling interferometer (UNI), a second-phase amplitude correction AO, and a final-stage nulling interferometer (coronagraph).

Fig. 2
Fig. 2

Schematic complex amplitude distributions in each stage of the high-contrast imaging system with UNI: (a) at the collimated beam of a telescope, (b) at one arm in the UNI after phase compensation by the first AO system, (c) at the other arm after a slight reduction of the unaberrated amplitude, (d) after the unbalanced nulling, (e) after the phase and amplitude compensation by the second PAC AO, and (f) after the last nulling interferometer (coronagraph).

Fig. 3
Fig. 3

Optical setup of a high-contrast imaging system experiment with the UNI and PAC AO as well as the 3D Sagnac nulling interferometric coronagraph.

Fig. 4
Fig. 4

Aberration magnification ratio for the amplitude difference g (cross) measured data, (solid line); M magnification ratio for two wavefronts without correlation (Eq. (9); and (dashed line) M 0.5 , with correlation of 0.5 (independency f = 0.5 ) in Eq. (11).

Fig. 5
Fig. 5

Measured complex amplitudes of arm 1 (square), arm 2 (star), after the UNI operation (triangle), and after the PAC AO operation (cross).

Fig. 6
Fig. 6

Rms of the aberrations in each stage. (a) Triangle: amplitude; (b) diamond: phase in radians; and (c) open circle: modulus of the NCA.

Fig. 7
Fig. 7

(Upper) Acquired images (using monochromatic 671 nm laser) of the focal plane camera (a) using the Arm 1 beam through the bright output of the 3D Sagnac interferometric coronagraph, (b) the Arm 1 beam through the nulled output of the 3D Sagnac, (c) the UNI output beam observed at the nulled output of the 3D Sagnac, and (d) after the PAC AO operation of (c). (Lower) Azimuthally averaged intensity profiles of the upper four images. The acquired images are normalized by the peak value of (a) “Arm 1 & Bright” output.

Fig. 8
Fig. 8

Comparison of the integrated speckle intensity through the 3D Sagnac nulling interferometer and σ ε 2 at each stage. (a) Integrated speckle intensity within the 2.7 λ / D radius; (b) variance of the CAC modulus σ ε 2 ; and (c) σ ε 2 with the NCP effect.

Tables (1)

Tables Icon

Table 1 Measured Amplitude Aberration, Phase Aberration, σ ε / E 0 , σ ε , σ ε 2 , σ ε 2 with NCP Aberrations (NCPA) and Integrated Speckle Intensity through 3D Sagnac Nulling Interferometer in Each Stage When g = 0.18 a

Equations (14)

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

E ( x , y ) = E 0 + ε ( x , y ) .
σ ε / E 0 = | ε / E 0 | 2 ,
σ ε 2 = | ε | 2 ,
{ E 1 = E A r m 0 + ε 1 E 2 = E A r m 0 + ε 2 ,
{ σ ε 1 / E A r m 0 = | ε 1 / E A r m 0 | 2 σ ε 1 2 = | ε 1 | 2 .
E U N I = E 1 ( 1 g ) E 2 = g E A r m 0 + [ ε 1 ( 1 g ) ε 2 ] ,
{ ε U N I = [ ε 1 ( 1 g ) ε 2 ] E U N I 0 = g E A r m 0 .
σ ε U N I / E U N I 0 = 2 ( 1 g ) + g 2 g σ ε 1 / E A r m 0 .
M = 2 ( 1 g ) + g 2 g .
σ ε U N I 2 = M 2 g 2 σ ε 1 2 .
M f = 2 ( 1 g ) f + g 2 g .
E P A C = E P A C + ε P A C = g E A r m 0 + h [ ε 1 ( 1 g ) ε 2 ] ,
{ σ ε P A C 2 = K 2 σ ε U N I 2 σ ε P A C / E P A C 0 = K σ ε U N I / E U N I 0 ,
σ ε P A C 2 / σ ε 1 2 g 2 ,

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