Instrumental stability requirements for exoplanet detection with a nulling interferometer: variability noise as a central issue.

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(lambda)) 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.

Full text

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Bruno Chazelas 9/2/2005 1/31

Instrumental Stability Requirements for Exoplanet Detection with a
Nulling Interferometer - Variability Noise is a Central Issue -

-------


Bruno Chazelas(1,*), Frank Brachet (1,**), Pascal Bordé(2,***), Bertrand Mennesson(3), Marc
Ollivier (1), Olivier Absil(4), Alain Labèque(1), Claude Valette(1), Alain Léger(1)

(1) Institut d’Astrophysique Spatiale, CNRS, bat. 121, Univ. Paris-Sud,
F-91405 Orsay, France,

(2) Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS19, Cambridge, MA
02138, USA

(3) Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive,
Pasadena CA 91109-8099 USA

(4) Institut d'Astrophysique et de Géophysique, Université de Liège, Allée du Six Août 17, B-
4000 Liège, Belgium



(*) PhD thesis co-funded by Alcatel Space Industry and CNRS
(**) PhD thesis co-funded by Alcatel Space Industry and CNES
(***) Michelson Postdoctoral Fellow

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Bruno Chazelas 9/2/2005 2/31




Running title: Variability Noise

Correspondence should be addressed to Alain.Leger@ias.u-psud.fr, tel: 33 1 69 85 85 80,
fax: 33 1 69 85 86 75

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Bruno Chazelas 9/2/2005 3/31

Abstract


We revisit the nulling interferometer performances that are needed for direct detection and
spectroscopic analysis of exoplanets, e.g. with the DARWIN(1) or TPF-I(2) missions. Two
types of requirements are found, one concerning the mean value of the instrumental nulling
function, <nl(λ)>, and another one regarding its stability. The former is usually pointed out. It
is stringent at short wavelengths but somewhat relaxed at longer wavelengths. The latter,
which we call Variability Noise condition, does not usually receive enough attention. It stands
whatever the telescope size and the stellar distance are. The results obtained by 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 mission required
performances. As already pointed out by Lay(3), this stability problem is not fully solved by
modulation techniques. Adequate solutions must be found that likely will include servo
systems using the stellar signal itself as a reference, and high stability internal metrology.



Copyright:

OCIS codes: 110.5100; 120.3180; 350.1260

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Bruno Chazelas 9/2/2005 4/31

1 Nulling Interferometers in the exoplanet research

The central problem of direct detection of extrasolar planets is the contrast between the star
and the planet (4 107 at 7 µm in the Sun/Earth case), and the need to cancel the stellar light
to analyze the planetary one.

In the IR spectral range, the law of diffraction imposes instruments of several tens of meters,
which, in space, can be reached only with diluted pupils. The principle of such an instrument,
a nulling interferometer, has been proposed as early as 1978 by Bracewell(4). The idea is to
have destructive interferences for the on-axis star, and constructive interferences for an off-
axis planet. The fraction of star light that is not cancelled out by the instrument is referred to
as the nulling ratio, often characterized by its inverse (the rejection factor).

Rejection factors expected for nulling interferometers (nullers) are of the order of 105,
significantly less than the star/planet contrast, a few 107 at 7 µm. Consequently, one needs
an additional separation of planet and star lights that uses subtraction techniques, and the
question of the instrumental stability must be carefully studied. As already stated by Lay(3) in
his analytical approach, the null stability drives most of the requirements on the instrument,
even if an internal modulation between different sub-interferometers is applied. This
statement is confirmed in the present paper and illustrated by means of laboratory
experiments.


2 Noises associated with stellar leakage

Different processes make the output signal of a nuller non-zero. Even with perfect optics, the
instrument’s transmission would be zero on-axis only, raising as θα off-axis (α = 2, 4,…,
according to the interferometer design), where θ is the angle between the axis and the

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Bruno Chazelas 9/2/2005 5/31
source. A stellar disk having a finite radius, the spatially integrated flux from the star is a
fraction, geom, of its total flux (Figure 1). In addition, the instrument is not perfect and
transmits a fraction, nl, of the on-axis flux. The stellar leakage generate a total photo-
electrons flux, at time t and wavelength λ:

Fsl (λ, t) = A.Fst(λ).[geom (λ, t) + nl(λ,t)], (1)
With :
A: a constant depending upon the telescope size, optics throughput, detector yield…;
Fst(λ): the incident stellar flux over a spectral bin centered around λ, in ph-el.m-2.s-1;
geom: the geometrical nulling ratio due to the finite size of the stellar disk and the non-flat
response of the interferometer around its axis (θα response), considering perfect optics;
geom << 1;
nl: the instrumental nulling ratio of the interferometer for an on-axis point source, taking into
account instrumental defects; nl << 1.
-----------
Figure 1
-----------
After an integration time τ, the number of photo-electrons due to stellar leakage is:

Nbsl(τ, λ) = A.Fst(λ).[<geom (λ)>_τ + <nl(λ)>_τ].τ, (2)

where <f>_τ represents the mean value, over the duration τ, of f:


We have






≡><
≡><
∫
∫
−
−
τ
τ
λ
(
ττλ
(
λ
(
ττλ
(
0
1
0
1
),_)
),_)
dttnlnl
dttgeomgeom
. (3)

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Bruno Chazelas 9/2/2005 6/31

For a constant stellar flux, both terms in sum (2) introduce noises due to the non-uniform rate
of photo-electron generation ("shot noise", also called "quantum noise" or, incorrectly,
"photon noise"), and to the stochastic variations of their mean values (their variability for
short). We will assume that the pointing of the interferometer towards the star is good
enough so that the variability noise of geom would be negligible with respect to its shot noise
(However the same analysis developed in this paper could be applied to this variability
noise). Because the variability affecting nl finds its origin in the instrument, we call it the
instrumental variability noise. It depends on the time variation of nl(t), or σ<nl>(τ), the
standard deviation of <nl>_τ. The total noise associated with nl is therefore the compound
effect of shot noise and instrumental variability. In this case, the photo-events form a doubly
stochastic Poisson process. Its variance is simply the sum of the shot noise and instrumental
variability variances(5).

The shot noise is proportional to the square root of the integrated flux, Nbsl
1/2, but the
variability noise is directly proportional to Fst(λ).

3 Requirement for the mean rejection factor, <nl(λ λ)>

If the interferometer is optimized, at a given wavelength, for a planet position with a relative
transmission of unity (planet on a bright fringe), after an integration τ, the signal due to the
planet is A Fpl τ. As the photon flux due to the planet is much weaker than that due to the
stellar leakage, the shot noise is given by the square root of the number of photo-electrons,
Nbsl. The Signal to Noise Ratio (SNR), if the shot noise were the only source of noise, would
be

S/Nsh = (A.τ)1/2.Fpl /[(Fst.(<geom> + <nl>)]1/2, (4)

which improves with integration time as τ1/2.

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Bruno Chazelas 9/2/2005 7/31
The shot noise associated with <nl> is instrument dependent, whereas that associated with
<geom> is intrinsic to an interferometer design and to the stellar angular size, and therefore
cannot be avoided. The maximum requirement for <nl> results: it should be somewhat
smaller than <geom>, so that the total shot noise would not be significantly increased by the
contribution of <nl>. The actual requirement is dictated by the total SNR needed at the
corresponding wavelength and depends upon the planetary signal intensity and the other
sources of noises, e.g. the Zodiacal light, the thermal background...

At a given wavelength, λ, the first transmission maximum of a Bracewell interferometer, with
base L, is at an angle θ = λ/(2L). This base can be selected so that the target planet is
located at one of the transmission maxima. The angular separation of a Sun-Earth system at
20 parsecs (65 light-years) is θ = 0.05 arcsec. When observing at 7µm, the planet will be
located on the first transmission maximum if the interferometer’s baseline is L = 14 m. The
geometrical stellar leakage due to the finite size of the stellar disk is then geom = 1.8 10-5.
The most interesting target planets are located in the Habitable Zone(6) (HZ) of their stars, i.e.
in the region where water can be liquid (roughly 0.7−1.5 AU for the Sun, i.e. zone between
Venus and Mars). Most stars in the DARWIN/TPF target lists are cooler than the Sun(7) and
have a larger ratio between their angular size and the distance to their HZ. They produce
larger leaks when a planet is searched for in their HZs. Therefore, the preceding value for
geom can be considered as the lowest one, and the condition derived thereafter for <nl> as
the most severe one.

At short wavelengths, i.e. around 7µm, the planetary signal is weak, whereas this is an
important domain to detect H2O and CH4 (Figure 2). Consequently, it is critical that the
instrument does not induce a significant additional noise contribution. A possible requirement
would be <nl> = 0.56 <geom>, which corresponds to an increase of the total shot noise by 25
%. For an interferometer with leakages similar to those of a Bracewell interferomter
(geom = 1.8 10-5 for an Earth-Sun like system), a sensible value for the mean instrumental
leakage would then be

<nl>(7µm) = 1.0 10-5. (5)

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Bruno Chazelas 9/2/2005 8/31

For another interferometer design with more intrinsic leakages, the requirement on <nl>
could be relaxed, but the required integration time, for a given telescope collecting area,
would increase due to the shot noise associated to the larger value of geom.
---------
Figure 2
---------

At long wavelengths, the requirement on <nl> can be relaxed because:
i. the Fstar/Fpl ratio diminishes, with an analytical upper estimate, in the 7-20µm domain
of [Fstar/Fpl](λ) < [Fstar/Fpl](7µm).[ λ/7µm]-3.37 (Figure 3);
ii. other sources of shot noise become important(1), e.g. the flux of the Local (Solar)
Zodiacal light, the thermal emission of optics...
---------
Figure 3
---------
The SNR due to the shot noise should have the same value as at 7µm. Considered alone,
point (i), leads to the condition:

<nl>/geom(λ) = [1.56 (Fpl(λ)/Fpl(7µm ))2 / (Fst(λ)/Fst(7µm))] – 1, (6)

which reduces to relation (5) when λ = 7µm and geom = 1.8 10-5.

To prepare the discussion of variability noise, it is convenient to use a more stringent
condition, <nl>/geom(λ) = 0.56 [(Fpl/Fst)(λ)]/[(Fpl/Fst)(7µm)], (Figure 4). This condition should
not be very difficult to achieve if that at 7µm is fulfilled. In the case of an optimized Bracewell
interferometer (geom = 1.8 10-5), it can be estimated as

<nl>(λ) = 1.0 10-5 (λ / 7µm)3.37. (7)

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Bruno Chazelas 9/2/2005 9/31

-------
Figure 4
-------

Remarks:
- Most of the stars, in the DARWIN/TPF-I target lists, are less luminous than the Sun
and correspond to a lower Fstar/Fpl ratio, for an Earth-size planet at T ∼ 300 K.
Consequently, the Sun/Earth case is the most demanding for the nulling instrument.

- We think that it is not possible to optimize the DARWIN/TPF-I instrument at 6 µm for
a 300 K planet, because the Fpl/Fst for such an object is then extremely low (Figure 3).
However, the performances of the nuller, at 6 µm should be kept as good as at 7 µm
(<nl>(6-7µm) = <nl>(7µm)) in order to be able to study somewhat warmer planets,
e.g. planets at 350 K.

- The above conclusion concerns exclusively shot noise, and holds even if the average
instrumental stellar leakage is removed by means of a modulation technique. This
contribution acts as a bias that cannot be calibrated analytically and hamper the
detection of the planet. If no subtraction/modulation technique were used, the null
depth needed for a detection should be significantly deeper than the planet/star
contrast of 1/ 4 107 = 2.5 10-8 at 7 µm, a very difficult goal indeed. In practice,
modulation techniques such as rotation of the array(4) (8) or phase chopping
techniques (9) (10) (11) (12) should remove this bias and allow a detection even if the mean
null is not so deep. However, these techniques are affected by null depth fluctuations
at all frequencies as proven by Lay(3): systematic errors are not completely removed
by modulation, and the actual requirements on the null will indeed be driven by the
requirements on the stability, i.e. variability noise, as discussed hereafter.

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Bruno Chazelas 9/2/2005 10/31
4 Requirements for Variability Noise
4.1 Signal to noise ratio due to Variability Noise

Assuming that the relative transmission of the instrument for the planetary signal is unity,
after integration during time τ, the signal is:

S = A.Fpl(λ).τ. (8)

Let us remind the reader that variability noise, Nv, is due to the stochastic variation of the
mean flux at the output of the (imperfect) nuller. It is proportional to the stellar flux and to the
standard deviation of the mean of the instrumental nulling ratio <nl> over the integration time
τ , σ<nl>_τ noted thereafter σ<nl>(τ), (see equation (3)).

Nv = A.Fst(λ).σ<nl>(τ).τ. (9)


The resulting SNR, is:

S/Nv = [Fpl /Fst (λ)].[1/σ<nl>(τ)]. (10)

The incidence of the integration time, τ, on the SNR through σ<nl>(τ) is a key element.
Conversely, if a minimum SNR is needed to obtain useful scientific information, a
requirement for σ<nl>(τ), results.


We can relate σ<nl>(τ) and the PSD of nl. We assume that nl is an ergodic random process
(ergodic random function of time), which means that its average and moments taken over the
time are the same as when taken over the different realizations. The output of an

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Bruno Chazelas 9/2/2005 11/31
instrumental nuller is a function of time having a finite duration T. From this output, we
estimate the statistical properties of the random process.

This random process has no finite energy in the sense of signal processing, ∫
∞
∞→
0
2
)( dtt nl
,
but has a finite power,
dttnlT
T
T
)( lim
2
0
1∫
−
∞→
, and a Power Spectral Density (PSD), noted
PSDnl.

2
2
0
1
)( lim)( dtet nlT PSD
ti
T
T nl
υπ
υ
−−
∞→
∫
≈

Such a PSD, when defined as a limit, follows the standard Fourier transform properties (for
details see Léna(13) and Goodman(5))

To simplify calculation we introduce the running average over τ, a random process noted
<nl>_τ(t) :


<nl>_τ(t) = τ-1 nl(t) ∗ ∏(t/τ),

where ∏(u) is the top-hat function (∏(u) = 1 for u between 0 and 1 and 0 elsewhere),
and ∗ represents the convolution. We have :

<nl>_τ(0) = <nl>_τ

As <nl>_τ(t) is a linearly filtered ergodic process it is also ergodic(5). Thus σ<nl>(τ), the
standard deviation of <nl>_τ for different outcomes of the random process, is equal to the
standard deviation of the random process <nl>_τ(t) over the time.

Using the Parseval theorem one shows(5) that

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Bruno Chazelas 9/2/2005 12/31

2
2
<
- )
υ
( = )(nl PSD
nl
∫
∞
∞−
>
τσ
_τ nl
.

As <nl>_τ has a non zero mean value, its PSD has a singular behavior in ν = 0. It goes
as <nl>2 δ(ν), where δ(ν) is the delta "function". This is because the Fourier transform at ν = 0
is the mean value of the function. The PSD' function, which is equal to PSD but in zero,
defined as:


)()(')('
2
__
υδυυ
ττττ
nlPSDPSD
−=
,

has a non-singular behavior in 0, and

σ<nl>
2
(τ) = PSD'nl _τ(υ)
−∞
∞∫



The Fourier transform of ∏(t) being sinc(ν) ≡ sin(πν)/πν, then σ<nl>(τ) derives from the Power
Spectral Density of the nulling function PSDnl :


d )
υ
(sinc )
υ
(' = )(
22
<
∫
∞
∞−
>
υττσ
nl
PSD
nl
(11)

If PSD'nl is decreasing rapidly on the scale ∆ν = τ-1, most of the contribution to the integral is
around zero. Then, sinc(ν) can be approximated by a top-hat function with FWHM = τ, and
equation (11) reads

d ) (' )(
/5 . 0
∫−
/5 . 0
2
<>
≈
τ
τ
υυτσ
nl
PSD
nl
(12)

If the fluctuations of nl are white, i.e., PSDnl(ν) is constant, the familiar result is found, σ<nl>(τ)
is proportional to τ-1/2 and S/Nv increases as τ1/2.

If there are instrumental drifts, the PSD of the nulling ratio will have a so-called “1/f”
component. This does not mean that PSDnl is diverging in 0. It means that the longer the time

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Bruno Chazelas 9/2/2005 13/31
interval between two calibrations of the instrument, the more noise power at low frequency.
Let T be the time separating two calibrations of the nuller. A possible model of this noise is :





≤+
>+
=
−
−−
1
11
for
for
)(
Tb aT
Tba
PSDnl
υ
υυ
υ


This model (Figure 5) gives a better representation of the physical reality, with a finite
energy for any finite value of T. Integrals in equation (12) or (11) are defined.

What is the effect of increasing the integration time τ, on the S/Nv ? In order to compare
different measurements with integration times τ, e.g. some with an interferometer orientation
yielding the planetary signal and other ones not doing so, the time T between two
calibrations, must be several times τ. Applying equation (12), one gets :


υ
d
υ
a
υ
d
υ
d
υ
(
τ
(
σ
ττ
bb aTT PSD
T
T
nlnl
)(2)(2);'2)
1
5 . 0
∫
0
5 . 0
∫
0
2
<
1
1
11
+++≈≈
−
>
∫
−
−
−−



ττ
2
τ
(
σ
bT
a
nl
++≈
><
))ln(1 (2)
2


If aτ >> b, the “1/f” noise is dominating the integral, and :

))
2
ln(1 (
a
2)(
2
<
τ
τσ
T
nl
+≈
>


which is very slowly decreasing with τ. The reduction of the integration domain is mostly
compensated by an increase of the PSD in the low frequency part. In presence of a
dominating 1/f noise, the S/N is almost independent of the integration time τ.

If aτ << b, white noise is dominating, and σ<nl>
2
(τ) ≈ b/τ . The familiar result is met, σ<nl>(τ) is
proportional to τ-1/2 and S/Nv increases as τ1/2.

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Bruno Chazelas 9/2/2005 14/31

4.2 Nulling experiments around the world

Considering a few nulling experiments presently performed around the world, is informative.
For several of them, the authors kindly provided us with the data files of their experimental
nl(t) function. We have computed the running average <nl>_τ(t) over τ, the PSD of nl and
the standard deviation for several integration times τ.

The results are shown in Figure 6, 7 and 8. For the best results (Figs.7 & 8) the mean values
miss the goal of 10-5 by a factor of 5 and 2, respectively. For τ =10s, they yield
σ<nl>(10s) ~ 10-6. At 7 µm, the Sun-to-Earth flux ratio, Fpl / Fst is 2.5 10-8. If this were the final
result, the S/Nv value resulting from (10) would be ∼ 2.5 10-2, whatever the stellar distance
and the telescope diameters, clearly an unacceptable situation.

-------
Figure 6,Figure 7,Figure 8
-------

Fortunately, longer integrations are expected to improve this SNR. If variability noise were
white, i.e. had a constant PSD, relation (12) predicts that, with increasing time, σ<nl>(τ) would
decrease as τ -1/2. Starting from the nuller performances reported for τ = 10 s, S/Nv would
increase to 2.5 10-2 (24 x 3600)1/2 ~ 7 at 7 µm, after a 10-day integration.

Unfortunately, all present experimental PSDs exhibit a 1/f-type peak at low frequencies.
Consequently, the standard deviations of nl decrease slower than τ -1/2 and extrapolation to
very long integration durations (days) does not look promising. These 1/f noises are probably
due to drifts in the experimental setups.

The issue of long-term drift is of major importance. Although this drift is difficult to control, the
situation is not hopeless. For instance, an important limitation for the instrumental nulling

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Bruno Chazelas 9/2/2005 15/31
ratio stems from the fine control of the Optical Path Difference (OPD) between the different
arms of the interferometer. Now, the servo adjustment of the delay line around a zero OPD
on DARWIN/TPF-I instruments will be performed using the stellar signal itself (fringe tracker
in the visible/near-IR) and should have no long-term drifts. However, an adequate stability
will be required for the instrument parts downstream the fringe tracker. The other limiting
factors of nl should be controlled as well, including intensity errors, pointing errors...
4.3 Required performances

When the noises associated with the stellar leakage (relation (1)) are dominant, e.g. at short
wavelengths, a possible noise budget is

Nv = Nsh = (1/2)0.5.Ntot.
(13)
To obtain spectroscopic information, at different wavelengths, including the most difficult
ones, the S/Ntot ratio must be sufficient in a maximum of 10 day integration, say S/Ntot ~ 7.
Relation (13) implies

S/Nv = (1/2)-0.5 S/Ntot ~ 10, in 10 days.

At 7µm, the star to planet contrast is 4 107 for the Earth/Sun case. At other wavelengths, the
relation Fst /Fpl(λ) can be deduced from Fig.3, or estimated from the (λ/7µm)-3.37 relation.
Equation (10) implies a requirement for the nuller stability that is shown in Fig.9. In the 7-
20µm domain, it can be written as:


σ<nl>(λ, 10days) ≤ 2.5 10-9 (λ / 7µm)3.37 (14)
or

σ<nl>(λ, 10 s) ≤ 7 10-7 (λ / 7µm)3.37 + white noise. (15)

----------
Figure 9
----------