Speckle level suppression using an unbalanced nulling interferometer in a high-contrast imaging system

Kaito Yokochi; Naoshi Murakami; Jun Nishikawa; Lyu Abe; Motohide Tamura; Alexander V. Tavrov; Mistuo Takeda; Takashi Kurokawa

doi:10.1364/OE.19.004957

Speckle level suppression using an unbalanced nulling interferometer in a high-contrast imaging system

Kaito Yokochi, Naoshi Murakami, Jun Nishikawa, Lyu Abe, Motohide Tamura, Alexander V. Tavrov, Mistuo Takeda, and Takashi Kurokawa

Optics Express
Vol. 19,
Issue 6,
pp. 4957-4969
(2011)
•https://doi.org/10.1364/OE.19.004957

Get Citation
Copy Citation Text
Kaito Yokochi, Naoshi Murakami, Jun Nishikawa, Lyu Abe, Motohide Tamura, Alexander V. Tavrov, Mistuo Takeda, and Takashi Kurokawa, "Speckle level suppression using an unbalanced nulling interferometer in a high-contrast imaging system," Opt. Express 19, 4957-4969 (2011)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (1040 KB)
Set citation alerts
Save article

Related Topics
Table of Contents Category
- Adaptive Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Active or adaptive optics (110.1080)
- Interferometry (120.3180)
- Astronomical optics (350.1260)
- Interference (260.3160)

About this Article
History
- Original Manuscript: January 4, 2011
- Revised Manuscript: February 9, 2011
- Manuscript Accepted: February 9, 2011
- Published: March 1, 2011

Accessible

Open Access

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.

Full Article | PDF Article

OSA Recommended Articles

Polarization interferometric nulling coronagraph for high-contrast imaging

Naoshi Murakami, Kaito Yokochi, Jun Nishikawa, Motohide Tamura, Takashi Kurokawa, Mitsuo Takeda, and Naoshi Baba
Appl. Opt. 49(16) D106-D114 (2010)

Exoplanet detection using a nulling interferometer

Manuel P. Cagigal and Vidal F. Canales
Opt. Express 9(1) 36-41 (2001)

On representing and correcting wavefront errors in high-contrast imaging systems

Amir Give'on, N. Jeremy Kasdin, Robert J. Vanderbei, and Yoav Avitzour
J. Opt. Soc. Am. A 23(5) 1063-1073 (2006)

References

View by:
Article Order
|
Year
|
Author
|
Publication

R. N. Bracewell, “Detecting nonsolar planets by spinning infrared interferometer,” Nature 274(5673), 780–781 (1978).
[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]
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]
M. Shao, J. K. Wallace, B. M. Levine, and D. Liu, “Visible nulling coronagraph,” Proc. SPIE 5487, 1296–1303 (2004).
[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]
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]
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]
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]
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]
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]
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]
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]
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 (3)

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]

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]

2009 (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]

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]

2008 (3)

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 (2)

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 (1)

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

2002 (1)

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 (1)

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 (1)

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 (1)

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]

Allard, F.

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]

Barman, T.

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.

Charton, J.

Chauvin, G.

Chiang, E.

Clampin, M.

Collins, B.

Doyon, R.

Egner, S.

Ehrenreich, D.

Feldt, M.

Fitzgerald, M. P.

Fusco, T.

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.

Give’on, A.

Goto, M.

Graham, J. R.

Gratadour, D.

Guyon, O.

Hashimoto, J.

Hayano, Y.

Henning, T.

Hodapp, K. W.

Janson, M.

Kalas, P.

Kandori, R.

Kasdin, N. J.

Kite, E. S.

Klahr, H.

Kotani, T.

Krist, J.

Kuchner, M. J.

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.

Kurokawa, T.

Kusakabe, N.

Lacombe, F.

Lafrenière, D.

Lagrange, A.-M.

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.

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.

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.

Montri, J.

Mordasini, C.

Morino, J.-I.

Mouillet, D.

Mugnier, L.

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]

Nishikawa, J.

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]

Patience, J.

Pluzhnik, E. A.

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.

Ridgway, S. T.

Rouan, D.

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]

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.

Stapelfeldt, K.

Suto, H.

Suzuki, R.

Takeda, M.

Tamura, M.

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]

Tavrov, A. V.

Thalmann, C.

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.

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]

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.

Appl. Opt. (1)

Astron. Astrophys. (3)

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]

Astron. Astrophys. Suppl. Ser. (1)

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. (3)

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. (1)

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

Nature (1)

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

Opt. Lett. (1)

Proc. SPIE (1)

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. (1)

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 (2)

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

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).

Download Full Size | PPT Slide | PDF

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).

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

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).

Download Full Size | PPT Slide | PDF

Fig. 5 Measured complex amplitudes of arm 1 (square), arm 2 (star), after the UNI operation (triangle), and after the PAC AO operation (cross).

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

Tables (1)

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

View Table

Equations (14)

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

E (x, y) = E^{0} + ε (x, y) .

σ_{ε / E^{0}} = \sqrt{〈 {| ε / E^{0} |}^{2} 〉},

σ_{ε}^{2} = 〈 {| ε |}^{2} 〉,

{\begin{cases} E_{1} = E_{A r m}^{0} + ε_{1} \\ E_{2} = E_{A r m}^{0} + ε_{2} \end{cases},

{\begin{matrix} σ_{ε_{1} / E_{A r m}^{0}} = \sqrt{〈 {| ε_{1} / E_{A r m}^{0} |}^{2} 〉} \\ σ_{ε_{1}}^{2} = 〈 {| ε_{1} |}^{2} 〉 \end{matrix} .

E_{U N I} = E_{1} - (1 - g) E_{2} = g E_{A r m}^{0} + [ε_{1} - (1 - g) ε_{2}],

{\begin{matrix} ε_{U N I} = [ε_{1} - (1 - g) ε_{2}] \\ E_{U N I}^{0} = g E_{A r m}^{0} \end{matrix} .

σ_{ε_{U N I} / E_{U N I}^{0}} = \frac{\sqrt{2 (1 - g) + g^{2}}}{g} σ_{ε_{1} / E_{A r m}^{0}} .

M = \frac{\sqrt{2 (1 - g) + g^{2}}}{g} .

σ_{ε_{U N I}}^{2} = M^{2} g^{2} σ_{ε_{1}}^{2} .

M_{f} = \frac{\sqrt{2 (1 - g) f + g^{2}}}{g} .

\begin{matrix} 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}], \end{matrix}

{\begin{matrix} 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}} \end{matrix},

σ_{ε_{P A C}}^{2} / σ_{ε_{1}}^{2} ≅ g^{2},

	Arm1	Arm2	UNI	PAC	$g \times$ PAC	$M_{c a l}$	$K_{c a l}$	S
Amp. aberration [%]	4.6	4.0	16	5.5	0.99	3.7	0.34	…
Phase aberration [λ ]	1/118	1/104	1/16	1/96	1/550	6.9	0.17	…
$σ_{ε / E^{0}}$	0.070	0.073	0.48	0.085	0.015	6.7	0.18	…
$σ_{ε}$ _[a.u.]	0.070	0.073	0.075	0.014	…	1.1	0.19	…
$σ_{ε}^{2}$ ( × 10⁻⁴) [a.u.]	49	53	56	1.8	…	1.1	0.032	0.035
$σ_{ε}^{2}$ (with NCPA) ( × 10⁻⁴) [a.u.]	100	110	69	3.2	…	0.65	0.046	0.030
Speckle ( × 10⁻⁴) [a.u.]	110	80	37	7.0	…	0.39	0.19	0.073