Abstract

We revisit the nulling interferometer performances that are needed for direct detection and the spectroscopic analysis of exoplanets, e.g., with the DARWIN [European Space Agency-SCI 12 (2000)] or TPF-I [JPL Publ. 05-5, (2005)] missions. Two types of requirement are found, one concerning the mean value of the instrumental nulling function nl(λ) and another regarding its stability. The stress is usually put on the former. It is stringent at short wavelengths but somewhat relaxed at longer wavelengths. The latter, which we call the variability noise condition, does not usually receive enough attention. It is required regardless of telescope size and stellar distance. The results from three nulling experiments performed in laboratories around the world are reported and compared with the requirements. All three exhibit 1/f noise that is incompatible with the performances required by the mission. As pointed out by Lay [Appl. Opt. 43, 6100–6123 (2004)], this stability problem is not fully solved by modulation techniques. Adequate solutions must be found that are likely to include servo systems using the stellar signal itself as a reference and internal metrology with high stability.

© 2006 Optical Society of America

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References

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  1. R. N. Bracewell, "Detection of nonsolar planets by spinning infrared interferometer," Nature (London) 274, 780-781 (1978).
    [CrossRef]
  2. O. P. Lay, "Systematic errors in nulling interferometers," Appl. Opt. 43, 6100-6123 (2004).
    [CrossRef] [PubMed]
  3. J. W. Goodman, Statistical Optics (Wiley Classics Library, 2000).
  4. J. F. Kasting, D. P. Whitmire, and R. T. Reynolds, "Habitable zones around main sequence stars," Icarus 101, 108-128 (1993).
    [CrossRef] [PubMed]
  5. A. Stankov, L. Kaltenegger, and C. Eiroa, "DARWIN star catalogue," European Space Agency/European Space Telecommunication internal publication DMS/SCI-A/DARWIN/208 (2005).
  6. M. Fridlund, "DARWIN, concept and feasibility study report," ESA-SCI 12 (European Space Agency, 2000), pp. 1-218.
  7. B. Mennesson and J-M. Mariotti, "Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets," Icarus 128, 202-212 (1997).
    [CrossRef]
  8. J. R. P. Angel and N. J. Woolf, "An imaging nulling interferometer to study extrasolar planets," Astrophys. J. 475, 373-379 (1997).
    [CrossRef]
  9. A. Karlsson and B. Mennesson, "The Robin Laurance nulling interferometers," in Astronomical Telescopes and Instrumentation 2000: High-Resolution Astronomy, Proc. SPIE 4006, 871-874 (2000).
    [CrossRef]
  10. O. Absil, A. Karlsson, and L. Kaltenegger, "Inherent modulation: a fast chopping method for nulling interferometry," in Astronomical Telescopes and Instrumentation 2002: High Resolution Astronomy, Proc. SPIE 4852, 431-435 (2003).
    [CrossRef]
  11. B. Mennesson, A. Léger, and M. Ollivier, "Direct detection and characterization of extrasolar planets: the Mariotti space interferometer," Icarus 178, 570-588 (2005).
    [CrossRef]
  12. P. Léna, F. Lebrun, and F. Mignard, Observational Astrophysics (Springer-Verlag, 1998), p. 453.
  13. M. Ollivier, "Contribution à la recherche d'exo-planètes. Coronographie interférentielle pour la mission DARWIN," Ph.D. dissertation (Université de Paris-Sud, Orsay, 1999).
  14. M. Ollivier, J.-M. Mariotti, A. Léger, P. Sekulic, J. Brunaud, and G. Michel, "Interferometric coronography for the DARWIN Space Mission--laboratory demonstration experiment," Astron. Astrophys. 370, 1128-1136 (2001).
    [CrossRef]
  15. R. Flatscher, Z. Sodnik, K. Ergenzinger, U. Johann, and R. Vink, "DARWIN nulling interferometer breadboard I: system engineering and measurement," M. Fridlund and T. Henning, eds., ESA Publ. SP-539, pp. 283-291.

2005

B. Mennesson, A. Léger, and M. Ollivier, "Direct detection and characterization of extrasolar planets: the Mariotti space interferometer," Icarus 178, 570-588 (2005).
[CrossRef]

2004

2003

O. Absil, A. Karlsson, and L. Kaltenegger, "Inherent modulation: a fast chopping method for nulling interferometry," in Astronomical Telescopes and Instrumentation 2002: High Resolution Astronomy, Proc. SPIE 4852, 431-435 (2003).
[CrossRef]

2001

M. Ollivier, J.-M. Mariotti, A. Léger, P. Sekulic, J. Brunaud, and G. Michel, "Interferometric coronography for the DARWIN Space Mission--laboratory demonstration experiment," Astron. Astrophys. 370, 1128-1136 (2001).
[CrossRef]

2000

A. Karlsson and B. Mennesson, "The Robin Laurance nulling interferometers," in Astronomical Telescopes and Instrumentation 2000: High-Resolution Astronomy, Proc. SPIE 4006, 871-874 (2000).
[CrossRef]

1997

B. Mennesson and J-M. Mariotti, "Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets," Icarus 128, 202-212 (1997).
[CrossRef]

J. R. P. Angel and N. J. Woolf, "An imaging nulling interferometer to study extrasolar planets," Astrophys. J. 475, 373-379 (1997).
[CrossRef]

1993

J. F. Kasting, D. P. Whitmire, and R. T. Reynolds, "Habitable zones around main sequence stars," Icarus 101, 108-128 (1993).
[CrossRef] [PubMed]

1978

R. N. Bracewell, "Detection of nonsolar planets by spinning infrared interferometer," Nature (London) 274, 780-781 (1978).
[CrossRef]

Absil, O.

O. Absil, A. Karlsson, and L. Kaltenegger, "Inherent modulation: a fast chopping method for nulling interferometry," in Astronomical Telescopes and Instrumentation 2002: High Resolution Astronomy, Proc. SPIE 4852, 431-435 (2003).
[CrossRef]

Angel, J. R. P.

J. R. P. Angel and N. J. Woolf, "An imaging nulling interferometer to study extrasolar planets," Astrophys. J. 475, 373-379 (1997).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, "Detection of nonsolar planets by spinning infrared interferometer," Nature (London) 274, 780-781 (1978).
[CrossRef]

Brunaud, J.

M. Ollivier, J.-M. Mariotti, A. Léger, P. Sekulic, J. Brunaud, and G. Michel, "Interferometric coronography for the DARWIN Space Mission--laboratory demonstration experiment," Astron. Astrophys. 370, 1128-1136 (2001).
[CrossRef]

Eiroa, C.

A. Stankov, L. Kaltenegger, and C. Eiroa, "DARWIN star catalogue," European Space Agency/European Space Telecommunication internal publication DMS/SCI-A/DARWIN/208 (2005).

Ergenzinger, K.

R. Flatscher, Z. Sodnik, K. Ergenzinger, U. Johann, and R. Vink, "DARWIN nulling interferometer breadboard I: system engineering and measurement," M. Fridlund and T. Henning, eds., ESA Publ. SP-539, pp. 283-291.

Flatscher, R.

R. Flatscher, Z. Sodnik, K. Ergenzinger, U. Johann, and R. Vink, "DARWIN nulling interferometer breadboard I: system engineering and measurement," M. Fridlund and T. Henning, eds., ESA Publ. SP-539, pp. 283-291.

Fridlund, M.

M. Fridlund, "DARWIN, concept and feasibility study report," ESA-SCI 12 (European Space Agency, 2000), pp. 1-218.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley Classics Library, 2000).

Johann, U.

R. Flatscher, Z. Sodnik, K. Ergenzinger, U. Johann, and R. Vink, "DARWIN nulling interferometer breadboard I: system engineering and measurement," M. Fridlund and T. Henning, eds., ESA Publ. SP-539, pp. 283-291.

Kaltenegger, L.

O. Absil, A. Karlsson, and L. Kaltenegger, "Inherent modulation: a fast chopping method for nulling interferometry," in Astronomical Telescopes and Instrumentation 2002: High Resolution Astronomy, Proc. SPIE 4852, 431-435 (2003).
[CrossRef]

A. Stankov, L. Kaltenegger, and C. Eiroa, "DARWIN star catalogue," European Space Agency/European Space Telecommunication internal publication DMS/SCI-A/DARWIN/208 (2005).

Karlsson, A.

O. Absil, A. Karlsson, and L. Kaltenegger, "Inherent modulation: a fast chopping method for nulling interferometry," in Astronomical Telescopes and Instrumentation 2002: High Resolution Astronomy, Proc. SPIE 4852, 431-435 (2003).
[CrossRef]

A. Karlsson and B. Mennesson, "The Robin Laurance nulling interferometers," in Astronomical Telescopes and Instrumentation 2000: High-Resolution Astronomy, Proc. SPIE 4006, 871-874 (2000).
[CrossRef]

Kasting, J. F.

J. F. Kasting, D. P. Whitmire, and R. T. Reynolds, "Habitable zones around main sequence stars," Icarus 101, 108-128 (1993).
[CrossRef] [PubMed]

Lay, O. P.

Lebrun, F.

P. Léna, F. Lebrun, and F. Mignard, Observational Astrophysics (Springer-Verlag, 1998), p. 453.

Léger, A.

B. Mennesson, A. Léger, and M. Ollivier, "Direct detection and characterization of extrasolar planets: the Mariotti space interferometer," Icarus 178, 570-588 (2005).
[CrossRef]

M. Ollivier, J.-M. Mariotti, A. Léger, P. Sekulic, J. Brunaud, and G. Michel, "Interferometric coronography for the DARWIN Space Mission--laboratory demonstration experiment," Astron. Astrophys. 370, 1128-1136 (2001).
[CrossRef]

Léna, P.

P. Léna, F. Lebrun, and F. Mignard, Observational Astrophysics (Springer-Verlag, 1998), p. 453.

Mariotti, J.-M.

M. Ollivier, J.-M. Mariotti, A. Léger, P. Sekulic, J. Brunaud, and G. Michel, "Interferometric coronography for the DARWIN Space Mission--laboratory demonstration experiment," Astron. Astrophys. 370, 1128-1136 (2001).
[CrossRef]

Mariotti, J-M.

B. Mennesson and J-M. Mariotti, "Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets," Icarus 128, 202-212 (1997).
[CrossRef]

Mennesson, B.

B. Mennesson, A. Léger, and M. Ollivier, "Direct detection and characterization of extrasolar planets: the Mariotti space interferometer," Icarus 178, 570-588 (2005).
[CrossRef]

A. Karlsson and B. Mennesson, "The Robin Laurance nulling interferometers," in Astronomical Telescopes and Instrumentation 2000: High-Resolution Astronomy, Proc. SPIE 4006, 871-874 (2000).
[CrossRef]

B. Mennesson and J-M. Mariotti, "Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets," Icarus 128, 202-212 (1997).
[CrossRef]

Michel, G.

M. Ollivier, J.-M. Mariotti, A. Léger, P. Sekulic, J. Brunaud, and G. Michel, "Interferometric coronography for the DARWIN Space Mission--laboratory demonstration experiment," Astron. Astrophys. 370, 1128-1136 (2001).
[CrossRef]

Mignard, F.

P. Léna, F. Lebrun, and F. Mignard, Observational Astrophysics (Springer-Verlag, 1998), p. 453.

Ollivier, M.

B. Mennesson, A. Léger, and M. Ollivier, "Direct detection and characterization of extrasolar planets: the Mariotti space interferometer," Icarus 178, 570-588 (2005).
[CrossRef]

M. Ollivier, J.-M. Mariotti, A. Léger, P. Sekulic, J. Brunaud, and G. Michel, "Interferometric coronography for the DARWIN Space Mission--laboratory demonstration experiment," Astron. Astrophys. 370, 1128-1136 (2001).
[CrossRef]

M. Ollivier, "Contribution à la recherche d'exo-planètes. Coronographie interférentielle pour la mission DARWIN," Ph.D. dissertation (Université de Paris-Sud, Orsay, 1999).

Reynolds, R. T.

J. F. Kasting, D. P. Whitmire, and R. T. Reynolds, "Habitable zones around main sequence stars," Icarus 101, 108-128 (1993).
[CrossRef] [PubMed]

Sekulic, P.

M. Ollivier, J.-M. Mariotti, A. Léger, P. Sekulic, J. Brunaud, and G. Michel, "Interferometric coronography for the DARWIN Space Mission--laboratory demonstration experiment," Astron. Astrophys. 370, 1128-1136 (2001).
[CrossRef]

Sodnik, Z.

R. Flatscher, Z. Sodnik, K. Ergenzinger, U. Johann, and R. Vink, "DARWIN nulling interferometer breadboard I: system engineering and measurement," M. Fridlund and T. Henning, eds., ESA Publ. SP-539, pp. 283-291.

Stankov, A.

A. Stankov, L. Kaltenegger, and C. Eiroa, "DARWIN star catalogue," European Space Agency/European Space Telecommunication internal publication DMS/SCI-A/DARWIN/208 (2005).

Vink, R.

R. Flatscher, Z. Sodnik, K. Ergenzinger, U. Johann, and R. Vink, "DARWIN nulling interferometer breadboard I: system engineering and measurement," M. Fridlund and T. Henning, eds., ESA Publ. SP-539, pp. 283-291.

Whitmire, D. P.

J. F. Kasting, D. P. Whitmire, and R. T. Reynolds, "Habitable zones around main sequence stars," Icarus 101, 108-128 (1993).
[CrossRef] [PubMed]

Woolf, N. J.

J. R. P. Angel and N. J. Woolf, "An imaging nulling interferometer to study extrasolar planets," Astrophys. J. 475, 373-379 (1997).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

M. Ollivier, J.-M. Mariotti, A. Léger, P. Sekulic, J. Brunaud, and G. Michel, "Interferometric coronography for the DARWIN Space Mission--laboratory demonstration experiment," Astron. Astrophys. 370, 1128-1136 (2001).
[CrossRef]

Astrophys. J.

J. R. P. Angel and N. J. Woolf, "An imaging nulling interferometer to study extrasolar planets," Astrophys. J. 475, 373-379 (1997).
[CrossRef]

Icarus

B. Mennesson and J-M. Mariotti, "Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets," Icarus 128, 202-212 (1997).
[CrossRef]

J. F. Kasting, D. P. Whitmire, and R. T. Reynolds, "Habitable zones around main sequence stars," Icarus 101, 108-128 (1993).
[CrossRef] [PubMed]

B. Mennesson, A. Léger, and M. Ollivier, "Direct detection and characterization of extrasolar planets: the Mariotti space interferometer," Icarus 178, 570-588 (2005).
[CrossRef]

Nature

R. N. Bracewell, "Detection of nonsolar planets by spinning infrared interferometer," Nature (London) 274, 780-781 (1978).
[CrossRef]

Proc. SPIE

A. Karlsson and B. Mennesson, "The Robin Laurance nulling interferometers," in Astronomical Telescopes and Instrumentation 2000: High-Resolution Astronomy, Proc. SPIE 4006, 871-874 (2000).
[CrossRef]

O. Absil, A. Karlsson, and L. Kaltenegger, "Inherent modulation: a fast chopping method for nulling interferometry," in Astronomical Telescopes and Instrumentation 2002: High Resolution Astronomy, Proc. SPIE 4852, 431-435 (2003).
[CrossRef]

Other

A. Stankov, L. Kaltenegger, and C. Eiroa, "DARWIN star catalogue," European Space Agency/European Space Telecommunication internal publication DMS/SCI-A/DARWIN/208 (2005).

M. Fridlund, "DARWIN, concept and feasibility study report," ESA-SCI 12 (European Space Agency, 2000), pp. 1-218.

J. W. Goodman, Statistical Optics (Wiley Classics Library, 2000).

P. Léna, F. Lebrun, and F. Mignard, Observational Astrophysics (Springer-Verlag, 1998), p. 453.

M. Ollivier, "Contribution à la recherche d'exo-planètes. Coronographie interférentielle pour la mission DARWIN," Ph.D. dissertation (Université de Paris-Sud, Orsay, 1999).

R. Flatscher, Z. Sodnik, K. Ergenzinger, U. Johann, and R. Vink, "DARWIN nulling interferometer breadboard I: system engineering and measurement," M. Fridlund and T. Henning, eds., ESA Publ. SP-539, pp. 283-291.

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Figures (9)

Fig. 1
Fig. 1

Transmission of a nulling interferometer with a θ 2 transmission around its axis. The finite size of the stellar disk leads to leaks with a relative intensity geom.

Fig. 2
Fig. 2

Spectrum of a planet with H 2 O and CH 4 in their atmospheres.[6] The 7 9 µm range is important for detecting these two important species.

Fig. 3
Fig. 3

Ratio of the Sun–Earth fluxes versus wavelength normalized at 7 µm . The fluxes are approximated by Planck functions at 5800 and 288 K , respectively. In the 7 20 µm domain the power law curve ( λ / 7 µm ) - 3.37 can be used as an upper value for this ratio.

Fig. 4
Fig. 4

Condition of n l ( λ ) imposed by the shot noise for a Bracewell interferometer ( geom = 1.8 × 10 - 5 ) . The condition is most severe at 7 µm , where we required that noise due to the instrumental stellar leakage not significantly increase noise due to the intrinsic stellar leakage geom. At longer wavelengths the actual requirement is that of the n l ( λ ) curve [Eq. (6)]. In preparing the discussion on the standard deviation of the variability noise (Section 4), it is convenient to impose the more stringent condition n l ( 7 µm ) ] [ ( F p l / F s t ) ( λ ) ] / [ ( F p l / F s t ) ( 7 µm ) ] or its analytical estimate n l ( 7 µm ) ] ( λ / 7 µm ) 3.37 .

Fig. 5
Fig. 5

Mathematical model used to describe the 1 / f noise. It has finite power, but this power grows with time T, the time separating two calibrations of the instrument.

Fig. 6
Fig. 6

Results from the nulling experiment by Ollivier and co-workers[13, 14] in 1999, using a CO 2 laser ( 10.6 µm ) . From top to bottom: (1) Experimental nulling function n l ( t ) versus time. (2) Running average over a 10 s duration. (3) PSD of the nulling function n l ( t ) . The increase in the PSD for low values of ν is clear. In the range of 0.3 0.01 Hz the PSD goes approximately as ν - 1.35 , a behavior close to the canonical 1 / f 1 / ν behavior. (4) Standard deviation of the mean value of n l over time τ, σ n l ( τ ) . In the frequency range investigated by that experiment, σ decreases with τ but more slowly than τ - 1 / 2 , which is typical of PSDs with 1 / f -like components.

Fig. 7
Fig. 7

Results from the nulling experiment by Alcatel (2004), using a laser diode at 1.57 µm and an integrated optics recombiner. From top to bottom: (1) Experimental nulling function versus time n l ( t ) . (2) Running average over 3 s duration. (3) The corresponding PSD. Its increase at low frequencies is clear. From 0.1 to 1 Hz it goes approximately as 1 / ν . (4) Standard deviation of the mean value of n l over time τ, σ n l ( τ ) . The behavior of σ with the integration time is similar to that in Fig. 6 and indicates that major improvements in the experimental setup are needed to obtain the required stability, i.e., a τ - 1 / 2 decrease with integration time (courtesy Alcatel Space Industry, 2004).

Fig. 8
Fig. 8

Quantities similar to those in Figs. 6 and 7 from the Astrium experiment in 2004.[15] The duration of the experiment is the longest of the three. The null curve between 1000 and 1500 s is selected to compute the PSD and n l standard deviation because it is the part with the best quality. A low-frequency increase in the PSD is present. From 0.01 to 0.1 Hz it goes approximately as 1 / ν . It must be emphasized that this experimental setup as well as the two preceding ones had the goal of a low null value but not that of its stability. Their authors have precise ideas about how to improve this stability (courtesy Astrium, Germany).

Fig. 9
Fig. 9

Required stability of the instrumental nulling function n l ( t ) on time scales τ = 10 s and 10 days. The levels achieved at shorter wavelengths in experiments reported in Figs. 7 and 8 are indicated.

Equations (233)

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

n l ( λ )
1 / f
4 × 10 7
7 µm
10 5
10 7
7 µm
θ α
α = 2 , 4 , ,
n l
F s l ( λ , t ) = A F s t ( λ ) [ geom ( λ , t ) + n l ( λ , t ) ] ,
F st ( λ )
m - 2 s - 1
θ α
geom 1
n l
n l 1
N b s l ( τ , λ ) = A F s t ( λ ) [ geom ( λ ) 〉̲ τ + n l ( λ ) 〉̲ τ ] τ ,
f 〉̲ τ
geom ( λ ) 〉̲ τ τ - 1 0 τ geom ( λ , t ) d t ,
n l ( λ ) 〉̲ τ τ - 1 0 τ n l ( λ , t ) d t .
n l
n l ( t )
σ n l ( τ )
n l 〉̲ τ
n l
N b s l 1 / 2
F s t ( λ )
n l ( λ )
A F p l τ
N b s l
S / N s h = ( A τ ) 1 / 2 F pl / [ F s t ( geom + n l ) ] 1 / 2 ,
τ 1 / 2
n l
geom
n l
geom
n l
θ = λ / ( 2 L )
20 pc
θ = 0.05 arc sec
7 µm
L = 14 m
geom = 1.8 × 10 - 5
0.7 1.5 AU
n l
7 µm
H 2 O
CH 4
n l = 0.56 geom
geom = 1.8 × 10 - 5
n l ( 7 µm ) = 1.0 × 10 - 5 .
n l
n l
F star / F p l
7 20 µm
( F star / F p l ) ( λ ) < ( F star / F p l ) ( 7 µm ) ( λ / 7 µm ) - 3.37
7 µm
n l / geom ( λ ) = { 1.56 [ F p l ( λ ) / F p l ( 7 µm ) ] 2 / [ F s t ( λ ) / F s t ( 7 µm ) ] } 1 ,
λ = 7 µm
geom = 1.8 × 10 - 5
n l / geom ( λ ) = 0.56 [ ( F p l / F s t ) ( λ ) ] / [ ( F p l / F s t ) ( 7 µm ) ]
7 µm
( geom = 1.8 × 10 - 5 )
n l ( λ ) = 1.0 × 10 - 5 ( λ / 7 µm ) 3.37 .
F star / F p l
T 300 K
6 µm
300 K
F p l / F s t
6 µm
7 µm
[ n l ( 6 7 µm ) = n l ( 7 µm ) ]
350 K
1 / 4 × 10 7 = 2.5 × 10 - 8
7 µm
S = A F p l ( λ ) τ .
N v
n l
σ n l 〉̲ τ
σ n l ( τ )
N v = A F s t ( λ ) σ n l ( τ ) τ .
S / N v = [ F p l / F s t ( λ ) ] [ 1 / σ n l ( τ ) ] .
σ n l ( τ )
σ n l ( τ )
σ n l ( τ )
n l
n l
0 n l 2 ( t ) d t ,
lim T T - 1 0 T n l 2 ( t ) d t ,
PSD n l
PSD n l ( υ ) lim T T - 1 | 0 T n l ( t ) exp ( 2 π i υ t ) d t | 2 .
n l 〉̲ τ ( t )
n l 〉̲ τ ( t ) = τ - 1 n l ( t ) Π ( t / τ ) ,
Π ( u )
Π ( u ) = 1
n l 〉̲ τ ( 0 ) = n l 〉̲ τ .
n l 〉̲ τ ( t )
σ n l ( τ )
n l 〉̲ τ
n l 〉̲ τ ( t )
σ n l 2 ( τ ) = PSD n l 〉̲ τ ( υ ) n l 2 .
n l 〉̲ τ
ν = 0
n l 2 δ ( ν )
δ ( ν )
ν = 0
PSD
PSD nl 〉̲ τ ( υ ) = PSD nl 〉̲ τ ( υ ) n l 2 ( υ )
σ n l 2 ( τ ) = PSD n l 〉̲ τ ( υ ) .
Π ( t )
sinc ( ν ) sin ( πν ) / πν
σ n l ( τ )
PSD n l
σ n l 2 ( τ ) = PSD n l ( υ ) sinc 2 ( τυ ) .
PSD n l
Δν = τ - 1
sinc ( ν )
σ n l 2 ( τ ) 0.5 / τ 0.5 / τ PSD n l ( υ ) .
n l
PSD n l ( ν )
σ n l ( τ )
τ - 1 / 2
S / N v
τ 1 / 2
1 / f
PSD n l
PSD n l ( υ ) = { a υ - 1 + b for | υ | > T - 1 a T + b for | υ | T - 1 .
S / N v
σ n l 2 ( τ ) 2 0 0.5 τ - 1 PSD n l ( υ ; T ) 2 0 T - 1 ( a T + b ) + 2 T - 1 0.5 τ - 1 ( a υ - 1 + b ) ,
σ n l 2 ( τ ) 2 a [ 1 + ln ( T 2 τ ) ] + b τ .
b
1 / f
σ n l 2 ( τ ) 2 a [ 1 + ln ( T 2 τ ) ] ,
1 / f
b
σ n l 2 ( τ ) b / τ
σ n l ( τ )
τ - 1 / 2
S / N v
τ 1 / 2
n l ( t )
n l 〉̲ τ ( t )
n l
10 - 5
τ = 10 s
σ n l ( 10 s ) 10 - 6
7 µm
F p l / F s t
2.5 × 10 - 8
S / N v
2.5 × 10 - 2
σ n l ( τ )
τ - 1 / 2
τ = 10 s
S / N v
2.5 × 10 - 2 ( 24 × 3600 ) 1 / 2 7
7 µm
1 / f
n l
τ - 1 / 2
1 / f
n l
N v = N s h = ( 1 / 2 ) 0.5 N tot .
S / N tot
S / N tot 7
S / N v = ( 1 / 2 ) - 0.5 S / N tot 10   in   10   days .
7 µm
4 × 10 7
F s t / F p l ( λ )
( λ / 7 µm ) - 3.37
7 20 µm
σ n l ( λ , 10   days ) 2.5 × 10 - 9 ( λ / 7 µm ) 3.37
σ n l ( λ , 10   s ) 7 × 10 - 7 ( λ / 7 µm ) 3.37 + white noise .
1 / f
τ - 1 / 2
n l = 10 - 5
7 µm
( σ n l / n l ) ( 10   days ) 2.5 × 10 - 4
( σ n l / n l ( 10   s ) 7 × 10 - 2 + white noise .
10 s
1 / f
1 / f
n l ( λ , t )
H 2 O
CH 4
7 20 µm
2 5
( 10 s )
CO 2
1.5 µm
( 6 20 µm )
θ 2
H 2 O
CH 4
7 9 µm
7 µm
288 K
7 20 µm
( λ / 7 µm ) - 3.37
n l ( λ )
( geom = 1.8 × 10 - 5 )
7 µm
n l ( λ )
n l ( 7 µm ) ] [ ( F p l / F s t ) ( λ ) ] / [ ( F p l / F s t ) ( 7 µm ) ]
n l ( 7 µm ) ] ( λ / 7 µm ) 3.37
1 / f
CO 2
( 10.6 µm )
n l ( t )
10 s
n l ( t )
0.3 0.01 Hz
ν - 1.35
1 / f 1 / ν
n l
σ n l ( τ )
τ - 1 / 2
1 / f
1.57 µm
n l ( t )
3 s
1 Hz
1 / ν
n l
σ n l ( τ )
τ - 1 / 2
1500 s
n l
0.1 Hz
1 / ν
n l ( t )
τ = 10 s

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