Chromatism compensation in wide-band nulling interferometry for exoplanet detection

Chromatism compensation in wide-band nulling

interferometry for exoplanet detection

Julien Spronck, Silvania F. Pereira, and Joseph J. M. Braat

We introduce the concept of chromatism compensation in nulling interferometry that enables a high rejection ratio in a wide spectral band. Therefore the achromaticity condition considered in most nulling interferometers can be relaxed. We show that this chromatism compensation cannot be applied to a two-beam nulling interferometer, and we make an analysis of the particular case of a three-telescope configuration. © 2006 Optical Society of America

OCIS codes: 120.3180, 350.1260.

1. Introduction

During the past decade, much effort has been dedi-cated to the search for exoplanets. The first exoplanet was discovered in 1995 by Mayor and Queloz by using

the radial velocity method.1 _{Since then, more than}

150 planets have been detected in less than 10 years. All of these planets were found by indirect methods, such as radial velocity measurements, astrometric

wobble, or photometric transits.2,3 _{However, until}

now, we have never detected direct radiation from an Earthlike planet. The challenge for direct detection of an Earthlike planet is the huge brightness contrast

between the star and the planet (106in the best case)

and the small angular separation (at a distance of 10 pc, the Sun–Earth distance is seen at an angle of

0.5␮rad). Promising techniques to meet this

chal-lenge are coronography4_{and nulling interferometry.}5

In this paper we concentrate on the latter.

Nulling interferometry consists in looking at a star–planet system with an array of telescopes, and then combining the light from these telescopes in such a way that, simultaneously, destructive inter-ference occurs for the starlight and (partially) con-structive interference for the planet light. The ratio between the intensities corresponding to constructive and destructive interference is called the rejection

ratio. To be able to detect a planet, this ratio should be of the order of at least 106_.

Reaching this ratio using monochromatic light is quite feasible theoretically, but the challenge is found

in achieving this over a wide spectral band (6–18␮m

or even wider6_{). This wide band is required to obtain}

spectral information from the planet and to optimally exploit the photon flux from the planet.

To reach a high rejection ratio in a wide band, most nulling interferometers use achromatic phase shift-ers (APS).7–10_{Indeed, it is usually thought that, if we}

want to work in a wide band, and if we want the same rejection ratio in that band, all components have to satisfy an achromaticity condition. In this paper we show that this is not always necessary.

In Section 2 we derive the general condition to have on-axis destructive interference for an array of N telescopes (nulling condition). We introduce a simple vectorial formalism to analyze this condition. In Sec-tion 3 we give a general expression for the rejecSec-tion ratio and we introduce the concept of chromatism compensation. In Section 4 we look at the conditions

to have a␪4_{-dependent transmission map. In Section}

5 we look at the simple case of a two-beam nulling interferometer, followed by the case of a three-beam nulling interferometer. In the latter, we apply the previously discussed concepts to the particular case where the phase shifters are delay lines, and we dem-onstrate how various parameters can be optimized if we want a better rejection ratio or a better sensitivity. Our conclusions are then summarized in Section 6.

2. Nulling Condition for an N-Telescope Array

In this section we derive the condition to have on-axis destructive interference for an array of N telescopes. The authors are with the Optics Research Group, Department of

Applied Sciences, Delft University of Technology, Lorentzweg 1, NL-2628 CJ Delft, The Netherlands. J. Spronck’s e-mail address is J.Spronck@tnw.tudelft.nl and J.Spronck@tudelft.nl.

Received 15 March 2005; accepted 11 July 2005. 0003-6935/06/040597-08$15.00/0

Let us consider N telescopes, all situated in the

plane z⫽ 0 and looking in the z direction (see Fig. 1).

The position of the jth telescope is given in polar

coordinates by共Lj,␦j兲. We assume that we can apply

independent phases and amplitudes␾jand Ajto each

beam before recombination. For a point source lo-cated at an angular separation from the optical axis ␪ and at an azimuth angle ␸, the detected complex amplitude f_␸共␪兲 is given by

f_␸(␪) ⫽

兺

j⫽1 N

Ajexp[ikLj␪ cos(␦j⫺ ␸)]exp(i␾j), (1)

where k is the wavenumber. Note that, if the obser-vation direction is different from the z axis, the opti-cal path lengths have to be compensated with delay lines.

We define the transmission map T_␸共␪兲 as the

nor-malized detected intensity,

T_␸(␪) ⫽ |f␸(␪)|

2

max

关

|f_␸(␪)|2

兴

. (2)

Since the angle␪ is small, we can expand the complex

amplitude around␪ ⫽ 0 according to

f_␸(␪) ⫽ f_␸(0)⫹ ␪f_␸⬘(0) ⫹␪

2

2 f␸⬙(0) ⫹ ␪3

6 f␸⵮(0) ⫹ · · · . (3)

To have on-axis共␪ ⫽ 0兲 destructive interference, we

must satisfy

f_␸(0)⫽

兺

j⫽1 N

Ajexp(i␾j)⫽ 0, (4)

and this can be decomposed into two conditions:

兺

j⫽1 N Ajcos␾j⫽ 0,

兺

j⫽1 N Ajsin␾j⫽ 0. (5)

If the complex amplitude from each telescope is

rep-resented by a vector with a length Ajand an angle␾j

with respect to a reference, the conditions in Eqs. (5) amount to nullify the sum of all vectors (see Fig. 2). Since we can choose a certain amplitude and a

certain phase as references, we actually have 共2N

⫺ 2兲 unknowns and only two conditions. Two of these unknowns can thus be determined as a function of the other unknowns. For example, let us assume that

these two unknowns are Al and Al⫹1. We can show

that we have

冉

Al Al⫹1

冊

⫽ 1 sin(␾l⫹1⫺ ␾l)

冉

sin␾l⫹1 ⫺cos ␾l⫹1 ⫺sin ␾l cos␾l

冊

⫻

冢

⫺j⫽1⫽l⫽l⫹1

兺

N Ajcos␾j ⫺

兺

j⫽1⫽l⫽l⫹1 N Ajsin␾j

冣

. (6)

By using these expressions to find the amplitudes Al

and Al⫹1, we are sure that the nulling conditions are

satisfied.

3. Rejection Ratio of an N-Telescope Array

If we define the rejection ratio as the ratio between the maximal and the minimal intensities of the in-terference pattern, we have that

R⫽ 1 T_␸(0) ⫽

冏

_j

兺

_⫽1N Aj

冏

2

冏

兺

_j⫽1N Ajexp(i␾j)

冏

2. (7)

The denominator of Eq. (7) is the squared modulus of the sum of all the vectors. If we want to reach a high rejection ratio in a wide spectral band, we actually have to satisfy the nulling conditions in Eqs. (5) for Fig. 1. Array of telescopes (dots) situated in the plane z_{⫽ 0 and}

looking in the z direction. The angles_{␪ and ␸ define the direction} of the incoming light. The position of the jth telescope is given in polar coordinates by_共Lj,␦j兲.

each wavelength in the spectral band. But, unlike what is usually thought, this does not imply that all the phases and all the amplitudes must be achro-matic. Each phase and amplitude could be wave-length dependent, provided that the sum of the vectors is equal to zero for every wavelength. Let us assume that we use a phase-matching device that will give chromatic phases. We usually think that this chromatism will limit the rejection ratio, but if we insert these chromatic phases in Eq. (6), we will find chromatic amplitudes Aland Al⫹1. There are thus

chromatic amplitudes that will compensate the phase-induced chromatism in such a way that the rejection ratio is not affected since the nulling condi-tions are satisfied in the whole spectral band (chro-matism compensation). Let us notice that the inverse is also possible. We could compensate the chromatism induced by a certain amplitude-matching device us-ing chromatic phases to fulfill the nullus-ing conditions for each wavelength present in the spectral band. There is thus, in nulling interferometry, a close rela-tion between amplitudes and phases.

Next we derive an expression for the rejection ratio with either amplitude or phase mismatching. Let us

first consider an amplitude mismatching ⑀m for the

mth beam, as shown vectorially in Fig. 3.

The amplitudes and the phases have been chosen so that the nulling conditions in Eqs. (5) are satisfied. Without mismatching, the sum is thus equal to zero. In presence of an amplitude mismatching, the mod-ulus of the sum is simply given by |⑀m|, in such a way

that the rejection ratio is given by

R⫽

冏

j

兺

⫽1 N Aj

冏

2 ⑀m 2 . (8)

We now assume a phase mismatching␦␾mfor the

mth beam, as shown in Fig. 4.

Similarly, the rejection ratio is given by

R⫽

冏

j

兺

⫽1 N Aj

冏

2 Am2␦␾m2 . (9)

For example, a phase mismatching of 1 mrad gives, for a two-beam nulling interferometer, a rejection ratio of R⫽ 4 ⫻ 106_.

4. ␪ Dependence of the Transmission Map of an

N-Telescope Array

A star is not a point source and has some nonnegli-gible finite size. For example, the angular diameter of our Sun seen from a distance of 10 pc is of the order of 5 nrad. To detect an exoplanet, we need not only a

high rejection ratio for ␪ ⫽ 0 but also for angular

separations␪ of a few nrad. The flatter the

transmis-sion map around␪ ⫽ 0, the easier it will be to reach

this extended rejection ratio. That is why a

transmis-sion map proportional to␪4 or, even better, to ␪6 is

preferred.

To have a␪4_{-transmission map, it follows from Eqs.}

(2) and (3) that, in addition to the nulling conditions, we must also satisfy

f_␸⬘(0) ⫽

兺

j⫽1 N

AjLjcos(␦j⫺ ␸)exp(i␾j)⫽ 0. (10)

Since this should be true for all azimuth angles, we actually have the following four conditions:

Q41⫽

兺

j⫽1 N AjLjcos␦jcos␾j⫽

兺

j⫽1 N AjXjcos␾j⫽ 0, (11a) Q42⫽

兺

j⫽1 N AjLjcos␦jsin␾j⫽

兺

j⫽1 N AjXjsin␾j⫽ 0, (11b) Q43⫽

兺

j⫽1 N AjLjsin␦jcos␾j⫽

兺

j⫽1 N AjYjcos␾j⫽ 0, (11c) Q44⫽

兺

j⫽1 N AjLjsin␦jsin␾j⫽

兺

j⫽1 N AjYjsin␾j⫽ 0. (11d) We can see that if the nulling conditions can be satisfied with chromatic phases and amplitudes, it is Fig. 3. Vectorial representation of the N beams with an

ampli-tude mismatching⑀mfor the mth beam.

not straightforward to fulfill these ␪4 _{conditions in}

Eqs. (11) chromatically since the positions of the tele-scopes are included in the conditions and obviously these cannot be wavelength dependent. Note that these conditions can only be satisfied if the number of telescopes is larger than two.

For use in a further example, let us rewrite the

expression of f⬘共0兲 using the definitions given in Eqs.

(11), leading to

f_␸⬘(0) ⫽ ik关(Q41cos␸ ⫹ Q43sin␸) ⫹ i(Q42cos␸

⫹ Q44sin␸)兴. (12)

5. Examples

In the previous sections we presented a general the-ory for an array of N telescopes. In reality, however,

the number of telescopes is limited to a few共N 艋 6兲.

Therefore it can be interesting to see how the nulling conditions in Eqs. (5) can be reached in the case of small N. Thus, in this section, we will look at two simple cases: a two- and a three-beam nulling inter-ferometer.

A. Two-Beam Nulling Interferometer

In this case we observe that satisfying the nulling conditions amounts to nullifying the sum of two vec-tors, as shown in Fig. 5.

It is obvious that the sum of these vectors can only be equal to zero if the vectors are opposite to each other, which means that the amplitudes of the two beams have to be the same, and that the phase

dif-ference between the two beams has to be equal to␲

for all wavelengths. For a two-beam nulling inter-ferometer, the use of an achromatic phase shifter

with␲ phase difference is necessary to reach a high

(theoretically infinite) rejection ratio in a wide spec-tral band.

B. Three-Beam Nulling Interferometer

For the case of three beams and considering the

sec-ond beam as a reference for the phases共␾2⫽ 0兲, we

have the following nulling conditions:

A1cos␾1⫹ A3cos␾3⫽ ⫺A2, (13a)

A1sin␾1⫹ A3sin␾3⫽ 0. (13b)

For the particular case where␾3共␭兲 ⫽ ⫺␾1共␭兲 ⫽ ␾␭,

we find

A3⫽ A1,

A2⫽ ⫺2A1cos␾␭, (14b)

which vectorially corresponds to Fig. 6.

This result is important in the sense that, for a three-beam nulling interferometer, it is possible to compensate the chromatism induced by the phase-matching device using chromatic amplitudes in such a way that the nulling conditions can be satisfied in a wide spectral band (chromatism compensation). In this case the use of achromatic phase shifters is not necessary once the amplitudes can be set according to Eqs. (14a) and (14b). The difficulty of having achro-matic phase shifters is then shifted to the difficulty of

obtaining accurate spectral profiles.11_{Note that}

chro-matism compensation is possible for every configura-tion with more than two telescopes.

1. Sensitivity of the Configuration

Here we will look at the mutual balance between amplitudes and phases in the particular case of a three-telescope interferometer. We will see that some configurations are well balanced while others are more critical. We consider further the monochromatic case of the configuration depicted in Fig. 6. If there is

a small phase mismatching␦␾ for the first beam (see

Fig. 7), it is possible to correct this phase mismatch-ing by changmismatch-ing the amplitudes of the second and the third beams. These corrections are given by

␦A2⫽ A2⫺ A2,0 A2,0 ⫽ ⫺␦␾ cot 2␾, (15a) ␦A3⫽ A3⫺ A3,0 A3,0 ⫽ ⫺␦␾ cot ␾. (15b) Although the parameters have a quite different nature, we can define a sensitivity parameter S as the ratio between the normalized amplitude correction and the phase mismatching:

S⫽␦A ␦␾ ⫽

冑

␦A2 2 ⫹ ␦A3 2 ␦␾ ⫽

冑

cot2␾ ⫹ cot22␾. (16)

Fig. 5. Vectorial representation of a two-beam nulling inter-ferometer.

Fig. 6. Vectorial representation of the studied three-beam nulling interferometer.

Lower values of S indicate a less-sensitive or better-balanced configuration. The minimum value

(see Fig. 8) is achieved approximately at␾ ⫽ 2␲兾3,

which corresponds to the configuration where the am-plitudes of the three beams are equal.

2. Optimal Constant Amplitude

In practice, we often use (quasi-)achromatic

amplitude-matching devices. Therefore it can be in-teresting to see what rejection ratio can be obtained with chromatic phases and constant amplitudes.

Hereafter all amplitudes are defined with respect

to the amplitude of the first beam A1. The term

con-stant amplitude means thus that the amplitudes A2

and A3will be of the type

Aj⫽ cjA1, (17)

where cjis a constant. To find the optimal values for

cj, we will consider A1to be independent of the

wave-length, but the results can be applied to the case

where A1is a function of wavelength as well. Because

of our choice of phase shifts关␾3共␭兲 ⫽ ⫺␾1共␭兲 ⫽ ␾␭兴, we

already know [see Eq. (14)] that A3 ⫽ A1, i.e., c3

⫽ 1.

We also know that there is a chromatic amplitude ratio a2⫽ A2兾A1for which the nulling conditions are

satisfied in the whole spectral band. This amplitude varies over the spectral band between a minimal

value amin and a maximal value amax. If we use a

constant amplitude a2⫽ c2, the rejection ratio will be

limited because the nulling conditions are not ful-filled for every wavelength. Defining the amplitude mismatching as the maximal distance between the used amplitude (constant in this case) and the nom-inal chromatic amplitude, this amplitude mismatch-ing will be minimal if

c2⫽

amin⫹ amax

2 . (18)

The amplitude A2,opt ⫽ c2A1 ⫽ A1共amin ⫹ amax兲兾2 is

called the optimal constant amplitude and we define

d as the corresponding amplitude mismatching.

Let us assume that, with a certain phase-matching

device, we want to reach a phase shift equal to ␣.

Because of this phase-matching device, we will

actu-ally have ␾_␭⫽ ␣ ⫹ ⑀共␭兲. There will thus be a phase

variation ⌬⑀ around the nominal phase shift ␣ (see

Fig. 9). Note that it is important to adjust the phase-matching device so that this phase variation is sym-metric around the desired phase shift. Indeed, this will lead to the minimal phase mismatching. Because

of this phase variation, the extremity of the vector A1

will describe an arc of a circle. The projection of this arc is equal to the distance d, previously defined as the amplitude mismatching corresponding to the op-timal constant amplitude.

To derive an expression for the rejection ratio, we choose the amplitude of the first beam as a reference

(A1⫽ A0for all wavelengths). Using Eq. (8), we find

that the minimal rejection ratio in the spectral band is given by

Rmin⫽

(2A0⫹ A2, opt) 2

d2 . (19)

If the phase variation ⌬⑀ does not depend on the

phase shift␣, it is geometrically obvious (see Fig. 9)

that the distance d will be minimal (the rejection

ratio will then be maximal) for␣ ⫽ ␲. But according

to Fig. 8, we see that the configuration corresponding

to ␣ ⫽ ␲ is very sensitive. In this case there is a

compromise between high rejection ratio and low sen-sitivity.

3. Delay Lines as Phase Shifters

If we use delay lines as phase shifters, then the phase shifts are of the type

␾k⫽

k␣ k0

, (20)

Fig. 8. Sensitivity parameter as a function of the phase shift_␾.

where k0is the reference wavenumber and is chosen

in such a way that the phase variation⌬⑀ is

symmet-ric around the phase shift␣. If kmand kMare

respec-tively the minimal and maximal wavenumbers, we can show that the reference wavenumber is given by

k0⫽

km⫹ kM

2 . (21)

If we define M as the ratio between the maximal and the minimal wavenumbers in the spectral band (also equal to the ratio between the maximal and the min-imal wavelengths), we can show that the phase

vari-ation⌬⑀ is given by

⌬⑀ ⫽ 2␣M_M⫺ 1_{⫹ 1}. (22)

In this case the phase variation depends on the phase

shift␣, but we can still show that the highest

rejec-tion ratio occurs when␣ ⫽ ␲. In this particular case

we have that

d⫽ 1 ⫹ cos

冉

2␲

M⫹ 1

冊

, A2,opt⫽ 1 ⫺ cos

冉

2␲

M⫹ 1

冊

. (23)

The minimal rejection ratio in the spectral band is then given by Rmin⫽

冏

3⫺ cos

冉

2␲ M⫹ 1

冊

冏

2

冏

1⫹ cos

冉

2␲ M⫹ 1

冊

冏

2. (24)

Obviously the rejection ratio decreases as the band-width increases, as shown in Fig. 10.

4. ␪ Dependence of the Transmission Map

To analyze the ␪ dependence of the transmission

map, we consider the case of delay lines as phase

shifters with␣ ⫽ ␲ in such a way that

␾1⫽

k␲ k0 ⫽ ⫺␾3

, ␾2⫽ 0. (25)

We assume that we have a chromatic amplitude-matching device in such a way that the amplitudes are given by

A3⫽ A1, A2⫽ ⫺ 2A1cos

k␲ k0

. (26)

With these amplitudes and phases, the four␪4

condi-tions in Eqs. (11) can be satisfied for only one

wave-length in the spectral band共k ⫽ k0兲 if the telescopes

are in a linear configuration:

X1⫽ ⫺L, X2⫽ 0, X3⫽ L, Y1⫽ Y2⫽ Y3⫽ 0. (27)

For other wavelengths, the␪4_{conditions are not}

ful-filled. We then have

Q41⫽ Q43⫽ Q44⫽ 0, Q42⫽ 2LA1sin

k␲ k0

. (28)

Replacing these expressions in Eq. (12) for␸ ⫽ 0, we

find |f '(0)|2_{⫽ 4k}2_L2_A 12sin2 k␲ k0 . (29)

Since this term is different from 0, the transmission

map will be dominated by the␪2 _{term, as shown in}

Subsection 5.B.5.

5. Numerical Example

Let us consider a spectral band from 500 to 650 nm.

In terms of wavenumbers, we have km ⫽ 9.67

⫻ 104_cm⫺1_{and k}

m⫽ 1.26 ⫻ 105cm⫺1, which gives the

value for M⫽ 1.3. Replacing these values in Eqs. (21),

(23), and (24), we find

k0⫽ 1.11 ⫻ 105cm⫺1, A2,opt⫽ 1.9172, R ⫽ 2238.

(30) The amplitude and the corresponding rejection ratio are plotted in Fig. 11.

Let us now assume that we apply the chromatic amplitude for which the nulling conditions are satis-fied in the whole spectral band. We consider the lin-ear configuration of Eq. (27) with a baseline of L ⫽ 75 m. After some calculations, using Eqs. (2), (3), and (29), and after integration over the spectral band, we find that the transmission map (or the null depth) is, for small␪共␪ ⬍ 10⫺7兲, given by

Tchrom(␪) ⬇ 1016␪2. (31)

Thus, if we want to reach a rejection ratio of 106 (a

null depth of 10⫺6), we must have␪10⫺6⬇ 10⫺2nrad.

Comparing this result to the achromatic case (see Fig. 12), we have that

Tachrom(␪) ⬇ 3.125 ⫻ 1034␪4)␪10⫺6⬇ 7.5 ⫻ 10⫺2nrad.

(32)

The rejection ratio of 106 can thus be reached for

angular separations 7.5 times larger with achromatic amplitudes and phases. Note that this number is independent of the baseline.

We can see from this example that the chromatism compensation can lead to a high rejection ratio in a wide spectral band, but unfortunately only for very small angular separations from the optical axis, which could lead to an important stellar leakage. Note that this is only an example, and there may be

configurations12 _{for which this effect is less}

impor-tant, or maybe the problem can be partially solved

using some internal modulation technique.13 _Note

also that we can use the chromatism compensation concept with other types of phase shifter for which the phases are less chromatic than with delay lines. With these phase shifters, this problem will be less important.

6. Conclusions

We have shown that the interferometric nulling con-ditions can be fulfilled in a wide spectral band, using chromatic phases and amplitudes, leading thus to a high rejection ratio in this band. Therefore we prove that an achromatic device is not always necessary. Chromatic devices can be used if we compensate the phase-induced chromatism by chromatic amplitudes or vice versa (chromatism compensation). There is thus a close relation between amplitudes and phases and the vectorial formalism is a simple and useful tool to look at this relation.

We have seen that, if the nulling conditions can be satisfied chromatically, it is not straightforward to

fulfill the ␪4 _{conditions with chromatic phases and}

amplitudes, which can lead to an important stellar leakage. However, there should be optimal configu-rations or modulation techniques for which this effect is reduced, but such configurations have not been studied yet and their existence has to be confirmed in the future.

We have also shown that chromatism compensa-tion is not possible for a two-beam nulling interferom-eter. For this kind of interferometer, it is thus mandatory to use achromatic phase shifters but also achromatic amplitude-matching devices and an ach-romatic beam combiner to reach a high rejection ratio in a wide spectral band.

We have looked in detail at the particular case of a three-beam nulling interferometer, where the phase shift between the second and the first beam is equal to the phase shift between the third and the second beam. We have seen that, with three beams, it is possible to use chromatism compensation and we Fig. 11. (a) Chromatic amplitude for which the rejection ratio is theoretically infinite for the whole spectral band (dashed– dotted curve) and optimal constant amplitude (solid curve). For this amplitude, the amplitude mismatching is equal to d. (b) Rejection ratio corre-sponding to the optimal constant amplitude. The minimal rejection ratio over the spectral band is equal to 2238.

have established an expression of the rejection ratio if we have a chromatic phase-shifting device and an achromatic amplitude-matching device. We have ap-plied this theory to the particular case where the phase shifters were delay lines. Finally, we have seen that for a certain spectral band, there are several parameters that we can use to optimize the rejection ratio (mean phase shift, reference wavenumber, and optimal amplitude). We have seen that optimizing the rejection ratio also leads to more sensitive config-urations. Using the approach presented in this paper, we have shown that a compromise has to be made between rejection ratio and sensitivity to detect an Earthlike exoplanet via nulling interferometry.

This research was performed at the Optics Research Group of Delft University of Technology and was sup-ported by the Knowledge Centre for Aperture Synthe-sis, a collaboration of the Nederlandse Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek and Delft University of Technology.

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