polarization nulling interferometry for exoplanet detection
TRANSCRIPT
Polarization nulling interferometry forexoplanet detection
Julien Spronck, Silvania F. Pereira and Joseph J.M. BraatOptics Research Group, Faculty of Applied Sciences,
Delft University of Technology,Lorentzweg 1, NL-2628 CJ Delft, The Netherlands
Abstract: We introduce a new concept of nulling interferometer withoutany achromatic device, using polarization properties of light. This typeof interferometer should enable a high rejection ratio in a theoreticallyunlimited spectral band. We analyze several consequences of the proposeddesign, notably, the possibility of fast internal modulation.
© 2006 Optical Society of America
OCIS codes: (120.3180) Interferometry; (260.5430) Polarization; (350.1260) Astronomicaloptics; (999.9999) Nulling interferometry.
References and links1. M. Mayor and D. Queloz, “A Jupiter-mass companion to a solar-type star,” Nature378, 355-359 (1995).2. N. Woolf and J. R. Angel, “Astronomical searches for earth-like planets and signs of life,” Astron. Astrophys.
36, 507-537 (1998).3. G. W. Marcy and R. P. Butler, “Detection of extrasolar giant planets,” Astron. Astrophys.36, 57-97 (1998).4. R. Bracewell, “Detecting nonsolar planets by spinning infrared interferometer,” Nature274, 780-781 (1978)5. J. R. Angel, A. Y. S. Cheng and N. J. Woolf, “A space telescope for IR spectroscopy of Earthlike planets,” Nature
232, 341-343 (1986)6. N. Baba and N. Murakami, “A method to image extrasolar planetswith polarized light,” Publ. Astron. Soc. Pac.
115, 1363-1366 (2003)7. J. Spronck, S. F. Pereira and J. J. M. Braat, “Chromatism compensation in wide-band nulling interferometry for
exoplanet detection,” Appl. Opt.45 (4), 597-604 (2006).8. R. M. A. Azzam and N. M. Bashara, “Ellipsometry and polarized light” (Elsevier, Amsterdam, 1987)9. P. Hariharan, “Achromatic and apochromatic halfwave and quarterwave retarders,” Opt. Eng.35 (11), 3335-3337
(1996).10. D. Mawet, J. Baudrand, C. Lenaerts, V. Moreau, P. Riaud, D. Rouan and J. Surdej “Birefringent achromatic
phase shifters for nulling interferometry and phase coronography,” Proceedings of Towards Other Earths: DAR-WIN/TPF and the Search for Extrasolar Terrestrial Planets, Heidelberg, Germany, 22-25 April 2003.
1. Introduction
The first exoplanet has been discovered in 1995 by Mayor and Queloz [1]. Since that moment,more than one hundred and fifty planets have been detected within ten years. All of them werefound by indirect methods [2, 3], which means that only some effects that the planet has on itsstar have been detected.
Direct detection of an Earth-like exoplanets is not an easy task. Indeed, if our solar systemwas seen from a distance of 10 pc, the angular separation between Earth and Sun would beequal to 0.5µrad and the brightness contrast between the star and the planet would be 106 at10 µm and significantly larger (1010) in the visible.
Nulling interferometry [4] seems a quite promising technique up to now. It consists in ob-serving a star-planet system with an array of telescopes, and then combining the light from
these telescopes in such a way that, simultaneously, destructive interference occurs for the starlight and (partially) constructive interference for the planet light. The ratio between the intensi-ties corresponding to constructive and destructive interference is called the rejection ratio. To beable to detect a planet, this ratio should be of the order of atleast 106, with the extra requirementthat it should be achieved in a wide spectral band (6–18µm or even wider [5]). Indeed, thiswide band is required to obtain spectral information from the planet and to optimally exploitthe photon flux from the planet.
To reach this high rejection ratio in a wide spectral band, most current nulling interferom-eters use a (achromatic) phase shifter. In this paper, we present a totally different approachthat makes use of the polarization properties of light, leading to a new way to achieve nullinginterferometry. Note that a similar analogy has been proposed in visible coronography [6].
In Section 2, we derive the generalized condition to have on-axis destructive interference foranN-telescope array, including the polarization effects. We apply this condition to a two- anda three-telescope configurations. In Section 3, we apply this concept in a wide spectral bandand we propose a design of a new type of nulling interferometer. In Section 4, we look at theinterference patterns that can be obtained with the proposed design. In Section 5, we establisha criterion to define an acceptable width of the spectral band. In Section 6, we look at thesensitivity of the proposed design with respect to some imperfections and misalignments. Ourconclusions are then summarized in Section 7.
2. Generalized nulling condition
In this section, we derive the general condition to have on-axis destructive interference for anarray ofN telescopes and we look at some implications of this condition, in the case of a two-and a three-beam nulling interferometer.
Let us consider an array ofN telescopes and let us assume that we can apply independentphases and amplitudesφ j andA j to each beam before recombination. To cancel the light fromthe star, we need on-axis destructive interference. We can show [7] that the condition to havesuch a destructive interference (nulling condition) is given by
N
∑j=1
A j exp(iφ j) = 0. (1)
We can divide both members of this equation by the factorA1exp(iφ1), in such a way that theamplitudesA j and the phasesφ j are defined relatively to the amplitude and phase of the firstbeam. Note that we assumed here that the relative amplitudesA j are not wavelength-dependentbut we did not make any assumption about the absolute spectraof the beams. Note also thatthese relative amplitudesA j and phasesφ j could be wavelength-dependent [7]. A more generalcondition can be derived assuming independent states of polarization for each beam. UsingJones formalism [8] to describe polarization, the generalized condition is given by
N
∑j=1
~A j exp(iφ j) =N
∑j=1
(
Ax, j
Ay, j
)
exp(iφ j) = 0, (2)
whereAx, j andAy, j are complex numbers. This generalized condition has very important con-sequences since it can lead to a new type of nulling interferometers, as discussed below.
2.1. Example 1: Two-beam nulling interferometer
In the case of a two-beam nulling interferometer, the generalized nulling condition in Eq. (2)simply amounts to
~A1exp(iφ1) = −~A2exp(iφ2) . (3)
In most current nulling interferometers, this condition issatisfied by applying aπ-phase shiftbetween the two beams (φ2 = φ1 + π). The condition in Eq. (3) could also be fulfilled withoutany phase shift but considering a polarization rotation ofπ (~A1 =−~A2). This is a fundamentallydifferent approach, as it will appear more clearly in the following example.
2.2. Example 2: Three-beam nulling interferometer
In this case, we have the following nulling condition.
~A1exp(iφ1)+ ~A2exp(iφ2)+ ~A3exp(iφ3) = 0. (4)
If we assume that all the beams have the same phase, we have
~A1 + ~A2 + ~A3 = 0. (5)
This condition can be fulfilled by rotating the polarizationof the beams. For example, if weimpose a horizontal linear state of polarization on the firstbeam, we could satisfy the conditionin Eq. (5) using
~A1 = A0
(
10
)
, (6a)
~A2 = A0
(
cos(
2π3
)
sin(
2π3
)
−sin(
2π3
)
cos(
2π3
)
)(
10
)
= A0
(
cos(
2π3
)
−sin(
2π3
)
)
, (6b)
~A3 = A0
(
cos(
4π3
)
sin(
4π3
)
−sin(
4π3
)
cos(
4π3
)
)(
10
)
= A0
(
cos(
4π3
)
−sin(
4π3
)
)
. (6c)
This shows that we can satisfy the nulling condition withoutany phase shifter by only rotatingthe polarization and consequently cancel the light coming from an on-axis star.
If a planet is orbiting around that star, then the planetary light coming from the differenttelescopes will have different optical path lengths. For that reason, it is interesting to look at thedetected intensity as a function of the optical path differences between the three beams. Let usfirst consider the monochromatic case. The detected amplitude as a function of the optical pathdifferences is given, within a phase factor, by
~Atot = ~A1 + ~A2exp(i2πOPD21/λ )+ ~A3exp(i2πOPD31/λ ) . (7)
whereOPD21 andOPD31 are respectively the optical path differences between beams 2 and 1and between beams 3 and 1. The detected intensity is then given by the square modulus of theamplitude in Eq. (7).
In the case of the example of Eqs. (6), we find the detected intensity depicted in Fig. 1.The rejection ratio, defined as the ratio between the maximaland minimal intensities of theinterference pattern, is theoretically infinite.
It is usually thought that beams with different coherent states of polarization cannot interferewith a high contrast. Our example shows that we can make threedifferently-polarized coherentbeams interfere with a theoretically perfect contrast. This is also true forN beams providedthat N > 2. The second consequence is that, since the intensity depends on the optical pathdifferences, it should be possible to have constructive interference for the light coming fromthe planet. The important fact is that the destructive interference takes place at the zero-OPDposition. In that case, there is no wavelength-dependent phase difference between the beams.
OPD21
(in µm)
OP
D31 (
in µ
m)
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 1. Normalized detected intensity (simulation) as a function of the optical path differ-ences (OPD) between the three beams.
3. Applications in wide-band nulling interferometry
In this section, we look at the generalized condition in a wide spectral band in order to designa new type of nulling interferometer.
To reach the amplitudes in Eqs. (6) for every wavelength in the spectral band, we would needachromatic polarization rotators. This is possible by combining different waveplates to createachromatic half-wave plates [9] or by using zero-order gratings [10]. But we choose a differentapproach, as explained below.
Suppose a system ofN beams, initially horizontally linearly polarized (x direction). Eachpolarization is then changed using a simple waveplate whoseprincipal axis makes an angleαwith the horizontal (see Fig. 2). IfTr andTα are the complex transmission coefficients of thewaveplate in its principal directions (Tr = |Tr| andTα = |Tα |exp(iφo−e), whereφo−e is the phasedifference between the ordinary and extraordinary axes), the Jones matrix of that waveplate atan angleα = 0 is given by
Mw(0) =
(
Tr 00 Tα
)
. (8)
For an orientation of the waveplateα, the Jones matrix would then be
Mw(α) = R(−α)Mw (0)R(α) =
(
Tr cos2 α +Tα sin2 α 12 sin2α (Tr −Tα)
12 sin2α (Tr −Tα) Tr sin2 α +Tα cos2 α
)
, (9)
whereR(α) is the rotation matrix at an angleα. The polarization state after the waveplate isthen given by
~A = A
(
Tr cos2 α +Tα sin2 α 12 sin2α (Tr −Tα)
12 sin2α (Tr −Tα) Tr sin2 α +Tα cos2 α
)(
10
)
= A
(
Tr cos2 α +Tα sin2 α12 sin2α (Tr −Tα)
)
.
(10)Since we want on-axis destructive interference without anyphase difference between the
beams and assuming that all the waveplates are exactly the same but with different orientations,
Fig. 2. The polarization of each beam (initially linear in thex direction (horizontal)) ischanged after a waveplate whose principal axis makes an angleα with the horizontal.
we must satisfy
N
∑j=1
A j
(
Tr cos2 α j +Tα sin2 α j12 sin2α j (Tr −Tα)
)
=
(
Tr ∑Nj=1 A j cos2 α j +Tα ∑N
j=1 A j sin2 α j12 (Tr −Tα)∑N
j=1 A j sin2α j
)
= 0 (11)
Using a simple waveplate in a wide spectral band,Tr andTα will be wavelength-dependentin such a way that the first component of the vector in Eq. (11) cannot be equal to zero for everywavelength. The second component, on the other hand, can be canceled achromatically by agood choice of the amplitudesA j and anglesα j. If we add, for each beam, a perfect verticallinear polarizer after the wave plate, the amplitude of thejth beam is then given by
~A j =
(
012A j (Tr −Tα)sin2α j
)
, (12)
and the nulling condition simply amounts to
N
∑j=1
A j sin2α j = 0. (13)
This condition is wavelength-independent. Therefore the null is achromatic if we assume iden-tical wavepelates.
In this proposed type of nulling interferometers, each beamencounters a horizontal linearpolarizer, a waveplate and a vertical linear polarizer (seeFig. 3). Thus, it should be possibleto reach a high rejection ratio in a wide spectral band with simple commercially availablecomponents. For example, in the case of a two-beam nulling interferometer and choosingA1 =A2, the condition to have an achromatic null is thatα2 = π −α1 or α2 = π/2+ α1. In orderto maximize the transmission of the interferometer, we can chooseα1 = π/4 andα2 = 3π/4.This can also be easily applied to a three-beam nulling interferometer by choosing for exampleA1 = A2 = A3, α1 = π/4, α2 = 7π/12 andα3 = 11π/12.
Note that we would have obtained similar results if the beamswere initially vertically linearlypolarized. We then could use a polarizing beam splitter instead of the first linear polarizer andapply the same principle to both outputs of the beam splitterin order to use the whole incomingintensity.
Fig. 3. Design of a new type of nulling interferometer, each beam encounters a horizontallinear polarizer, a waveplate and a vertical linear polarizer.
4. Transmission and modulation
In this section, we look at several direct consequences of the proposed design in the case of athree-beam nulling interferometer.
4.1. Transmission map
Let us considerN coplanar telescopes looking in the same directionz (see Fig. 4). The positionof the jth telescope is given in polar coordinates by(L j,δ j). For a point source located at anangular separation from the optical axisθ and at an azimuth angleϕ, the detected complexamplitude~fϕ (θ) is given by
~fϕ (θ) =N
∑j=1
~A j exp(ikL jθ cos(δ j −ϕ))
=N
∑j=1
(
012A j (Tr −Tα)sin2α j
)
exp(ikL jθ cos(δ j −ϕ)).
(14)
Note that this expression is not general in the sense that thestar which we point at lies onthe z-axis. If this was not the case, there would be additional delays that are not taken intoaccount here. Note also that this reasoning is valid for a space mission where longitudinaldispersion can be neglected. For any ground-based interferometer, this effect would decreasethe performances of the interferometer. In principle, the function~fϕ (θ) should also include thepolarization and birefringent properties of the individual telescopes. We have not included thiseffect here because we suppose that the telescopes behave identically in this respect and that,on recombination, these effects are canceled.
If we define the transmission mapTϕ (θ) as the normalized detected intensity, we have
Tϕ (θ) =
∣
∣
∣
~fϕ (θ)∣
∣
∣
2
max
[
∣
∣
∣
~fϕ (θ)∣
∣
∣
2] . (15)
4.2. θ -dependence of the transmission map
A star is not a point source but has some non-negligible finitesize. For example, the angulardiameter of our sun, seen from a distance of 10 pc, is of the order of 5 nrad. To detect an
q
j
(L2 cosd2 ,L 2 sin d2 ,0)
(LNcosdN ,LNsin dN,0)(L1 cosd1 ,L 1 sin d1 ,0)
z
x
y
Fig. 4. Array of telescopes (dots) situated in the planez = 0 and looking in thez direction.The anglesθ and ϕ define the direction of the incoming light. The position of thejth
telescope is given in polar coordinates by(
L j,δ j)
.
exoplanet, we need not only a high rejection ratio forθ = 0 but also for angular separationsθof a few nrad. The flatter the transmission map aroundθ = 0, the easier it will be to reach this“extended” rejection ratio. That is why a transmission map proportional toθ 4 or, even better,to θ 6 is preferred.
We can show [7] that, in order to have aθ 4-transmission map, we must satisfy, in addition tothe nulling condition in Eq. (13),
N
∑j=1
A j sin2α jL j cos(δ j −ϕ) = 0. (16)
Since this condition should be fulfilled for all anglesϕ, Eq. (16) can be split into two differentconditions
N
∑j=1
A j sin2α jL j cosδ j = 0, (17a)
N
∑j=1
A j sin2α jL j sinδ j = 0. (17b)
These conditions are different from theθ 4-conditions for other types of nulling interferom-eters [7]. However, we can show that, in the case of a three-telescope configuration, the onlypossible configuration to fulfill these conditions is a linear configuration, as it is the case forother nulling interferometers. For exoplanet detection, alinear configuration is less interest-ing because it only gives information in one direction. A solution to this lack of informationwould be to rotate the whole array of telescopes but this would give rise to slow modulation, asexplained in the next section. We can then conclude, that no interesting three-telescope config-uration can fulfill theθ 4-conditions.
4.3. Modulation
Another difficulty could prevent us from directly detectingan Earthlike exoplanet: the possibleemission from exo-zodiacal dust near the orbital plane of the planet, as in our own solar system.We, a priori, do not know anything about the exo-zodiacal cloud, but we can assume that it iscentro-symmetric. Because of this central symmetry, this problem could be handled by using
modulation techniques. A possible solution is to use external modulation, which consists inrotating the whole telescope array around its center, but this gives rise to very slow modulationand it will considerably decrease the number of targets thatwe can observe during a spacemission. A more convenient solution is internal modulation. With this technique, we do notchange the positions of the telescopes. Via optical means, we create different transmissionmaps that we combine in order to create modulation maps.
By changing the angleα j, we can change the “weight” of the amplitudeA j. Thus, we canthen change the ratio between the amplitudes of the different beams by simply rotating thewaveplates, provided that the nulling condition in Eq. (13)is satisfied. This has two conse-quences: the first one is that, with this type of nulling interferometer, we do not need anyextra amplitude-matching device, as it is the case in most ofcurrent nulling interferometers.The amplitude-matching is inherent to the design and is simply produced by a rotation of thewaveplate; the second and much more important consequence is that, since we can change theratio between the amplitudes of the beams, we can have a continuous set of transmission maps,which could be used for fast modulation.
Fig. 5 shows an example of a set of six transmission maps in thethree-telescope case thathave been obtained by only rotating the waveplates. In thesetransmission maps, the maximal
intensity has been normalized to a value given by(
∑Nj=1
∣
∣A j/A1sin2α j∣
∣
)2. Note that this is
just an example, out of a continuous range of transmission maps. Nevertheless, it can be shownthat any of these transmission maps can be represented by a linear combination of three others.Therefore, three different transmission maps are sufficient to get the whole continuous set (forexample, Fig. 5(a), 5(c) and 5(e) or 5(b), 5(d) and 5(f)). We also have to think, in more details,about a modulation strategy between these transmission maps.
5. Spectral response
In some applications, besides the detection of an Earthlikeexoplanet, spectral information ofthe light coming from the planet is needed in order to study its atmosphere. In this case, a widespectral band is required.
In the proposed design, if we make the assumption of perfect polarizers and exactly identi-cal waveplates, there is absolutely nothing in the nulling condition in Eq. (13) that limits thespectral band, so a high rejection ratio in an infinitely-wide spectral band is not unthinkable.However, in practice, this is not true since polarizers and waveplates are not perfect and arespectrally limited. Furthermore, as we will see in this section, the response of the interferome-ter is not the same for all wavelengths, that is, the detectedintensity is wavelength-dependent.
If we assume identical waveplates for each beam, the detected intensity will be proportionalto
I ∝ |Tr −Tα |2 , (18)
independently of the optical path length differences between the beams. The intensity for theconstructive interference is then also proportional to Eq.(18), which, in the case of a perfectwave plate is proportional to
I ∝ |1−exp(i∆φ)|2 = 4sin2 ∆φ2
, (19)
where∆φ is the phase difference between the two states of polarization induced by the wave-plate. If we furthermore consider conventional waveplates(as opposed to achromatic wave-plates), we have
∆φ =2πλ
(ne (λ )−no (λ ))d =2πλ
B(λ ) , (20)
y
xA1
2a2
A2
2a3
A3
y
x
A12a2A2
2a3
A3
2a1
y
x
A1
A2
2a3
A3
2a1
2a2
θx (in rad)
θ y (
in r
ad
)
−5 0 5
x 10−8
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10−8
0
0.5
1
1.5
2
2.5
3
θx (in rad)
θ y (
in r
ad
)
−5 0 5
x 10−8
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10−8
0
0.5
1
1.5
2
2.5
3
3.5
θx (in rad)
θ y (
in r
ad
)
−5 0 5
x 10−8
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10−8
0
0.5
1
1.5
2
2.5
3
(a) (b) (c)
y
x
A1
2a2
A2 2a3A3
2a1
y
x
A1
2a2
A2
A3
2a1
y
x
A1
2a2
A2
2a3
A3
2a1
θx (in rad)
θ y (
in r
ad
)
−5 0 5
x 10−8
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10−8
0
0.5
1
1.5
2
2.5
3
3.5
θx (in rad)
θ y (
in r
ad
)
−5 0 5
x 10−8
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10−8
0
0.5
1
1.5
2
2.5
3
θx (in rad)
θ y (
in r
ad
)
−5 0 5
x 10−8
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10−8
0
0.5
1
1.5
2
2.5
3
3.5
(d) (e) (f)
Fig. 5. Simulated three-telescope transmission maps corresponding to different waveplateorientations. All these maps have been calculated with the following parameters: A1 =A2 = A3, L1 = L2 = L3 = 25m andδ1 = 0,δ2 = 2π/3,δ3 = 4π/3, and for a spectral bandgoing from 500 to 650nm. (a) 2α1 = 0,2α2 = 2π/3,2α3 = 4π/3, (b) 2α1 = π/6,2α2 =π/6+2π/3,2α3 = π/6+4π/3, (c) 2α1 = 2π/6,2α2 = 2π/6+2π/3,2α3 = 2π/6+4π/3,(d) 2α1 = 3π/6,2α2 = 3π/6+2π/3,2α3 = 3π/6+4π/3, (e) 2α1 = 4π/6,2α2 = 4π/6+2π/3,2α3 = 4π/6+4π/3, (f) 2α1 = 5π/6,2α2 = 5π/6+2π/3,2α3 = 5π/6+4π/3
whereλ is the wavelength,ne (λ ) and no (λ ) are the extraordinary and ordinary refractiveindices,d is the thickness andB(λ ) is the birefringence of the waveplate.
The intensity is then maximum for∆φ = (2n+1)π (half-wave plate) and equal to zero for∆φ = 2nπ, wheren is an integer. This shows that some wavelengths will be well transmitted,while others will not be transmitted at all.
The criterion that we chose to define theacceptable spectral band is then that all the wave-lengths should be transmitted with at least half the maximalintensity, which leads to the fol-lowing condition
(4n+1)π2≤ ∆φ =
2πλ
B(λ ) ≤ (4n+3)π2
, (21)
We assume that the birefringence is constant in the spectralband. This is not a very realisticassumption but we can show, in the example of quartz, that it does not drastically affect thecriterion. In Fig. 6, we compare the spectral response of theinterferometer in the visible domainin the case of quartz waveplates and in the case of constant-birefringence approximation. Wecan see that the approximation is not very good but in both cases, the acceptable spectral bandis of the same order of magnitude. Furthermore, the birefringence is chosen in such a way thatthe waveplate is a half-wave plate for the wavelengthλ0. We then have
B = (2n+1)λ0
2. (22)
The minimal and maximal wavelengths in the acceptable spectral band are then given by
λmin =4n+24n+3
λ0 and λmax =4n+24n+1
λ0. (23)
We can characterize the bandwidth by defining
M =λmax
λmin=
4n+34n+1
. (24)
We can see that the bandwidth is maximum if we use zero-order waveplates (n = 0), whichin this case givesM = 3. For example, in the infrared region, this technique wouldallow usto work from 6 to 18µm. The spectral band will then probably be limited by the polarizers.Obviously, the acceptable spectral band can be wider if we use achromatic waveplates (see Fig.6).
Note that the presence of polarizers does not necessarily imply losses if we use achromaticwaveplates and a polarizing beam splitter to separate the two orthogonal states of polarizationand use both of them in a similar set-up.
6. Sensitivity to imperfections and misalignments
In this section, we will see how sensitive the proposed set-up is with respect to misalignmentsand imperfections and how these defects affect the rejection ratio.
Let us first derive a general expression for the on-axis detected intensity. We assume thateach polarizer could have its own imperfection, that is its own extinction ratio,εk, j, k = 1 or 2(εk, j represents the imperfection of the first or second polarizerfor the jth beam) and that eachwaveplate could be different for each beam (Tr, j andTα, j). The amplitude of thejth beam isgiven by
~A j = A j
(
ε2, j 00 1
)(
Tr, j cos2 α j +Tα, j sin2 α j12 sin2α j (Tr, j −Tα, j)
12 sin2α j (Tr, j −Tα, j) Tr, j sin2 α j +Tα, j cos2 α j
)(
1ε1, j
)
.
(25)
3 4 5 6 7 8 9
x 10−7
0
0.5
1
1.5
2
2.5
3
3.5
4
Wavelength (in m)
Inte
nsi
ty
n = 0, B = BQuartz
(λ)
n = 1, B = BQuartz
(λ)
n = 2, B = BQuartz
(λ)
n = 0, B = B0
n = 1, B = B0
n = 2, B = B0
Achromatic waveplate
Fig. 6. Spectral response of the interferometer in the visible domain in the case of quartzwaveplates (dash-dot lines) and in the case of constant-birefringenceapproximation (solidlines). We also compare zeroth-order (blue lines), first-order (green lines) and second-order(red lines) waveplates. The magenta solid line represents the spectral response in the caseof an achromatic waveplate made of a combination of quartz and magnesium fluoride.
After some calculations, we find that the on-axis detected intensity is given by
Iλ =∣
∣
∣∑Nj=1 A jε2, j
[
Tr, j cos2 α j +Tα, j sin2 α j +12ε1, j sin2α j (Tr, j −Tα, j)
]
∣
∣
∣
2
+∣
∣
∣∑Nj=1 A j
[
12 sin2α j (Tr, j −Tα, j)+ ε1, j
(
Tr, j sin2 α j +Tα, j cos2 α j)]
∣
∣
∣
2.
(26)
Hereafter, we will analyze each defect separately. For all the simulations, we used the fol-lowing parameters for the perfect case
A1 = A2 = A3 = 1,2α1 = 7π
6 ,2α2 = 11π6 ,2α3 = π
2 ,ε1,1 = ε1,2 = ε1,3 = 0,ε2,1 = ε2,2 = ε2,3 = 0,
Tr,1 = Tr,2 = Tr,3 = Tr = 1,Tα,1 = Tα,2 = Tα,3 = Tα = exp(i5πλ0/λ ) ,λmin = 500nm,λmax = 650nm,λ0 = 562nm.
(27)
6.1. Amplitude and phase mismatchings
Let us first consider an amplitude mismatchingAl = Al,0 + δAl . We can show that the on-axisdetected intensity is then given by
Iλ = |δAl sin2θl |2 (Tr −Tα)2
4∝ |δAl |
2 . (28)
The rejection ratio obtained with an amplitude mismatchingfor the first beam is depicted inFig. 7. The amplitude mismatching should be of the order of 10−3 or lower in order to have arejection ratio of 106.
If we consider now a phase mismatchingAl = Al,0exp(iδφl), we will find the same expres-sion for the on-axis intensity,
Iλ = |δφl sin2θl |2 (Tr −Tα)2
4∝ |δφl |
2 . (29)
We can then draw the same conclusion for the rejection ratio (not depicted here since it isidentical to the rejection ratio obtained with an amplitudemismatching). The phase mismatch-ing should be lower than 10−3, which is easily obtained in the infrared.
0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10−3
105
106
107
108
Amplitude mismatching, δAl
Reje
ctio
n r
ati
o
Fig. 7. Rejection ratio as a function of an amplitude-mismatchingδA1
6.2. Imperfections of the polarizers
Imperfections of the polarizers are modeled by settingε1, j andε2, j different from zero. Thesenumbers can be complex, giving rise to a certain ellipticityin the polarization state.
We chose all theε1, j andε2, j in such a way that their moduli are a random number between 0and 10−3 and the phases are also randomly chosen. We then look at the average rejection ratiothat can be obtained. The results are plotted in Fig. 8.
The rejection ratio is in average slightly higher than 106. The “amplitude imperfections” ofthe polarizers should then also be of the order of 10−3, which is not easy to satisfy in a widespectral band. Note that this requirement is less stringentif we use achromatic waveplates in-stead of conventional waveplates. Indeed, if we consider identical quasi-achromatic waveplates(Tα,1 = Tα,2 = Tα,3 = Tα = exp(i(π +δ )), whereδ << π) and identical imperfect polarizers(ε1,1 = ε1,2 = ε1,3 = ε2,1 = ε2,2 = ε2,3 = ε << 1), we can show that the on-axis intensity is thenproportional to
I ∝ (εδ )2 , (30)
which shows that imperfections of the polarizers can be compensated by very achromatic wave-plates and inversely, chromatic waveplates can be used combined with very good polarizers.
6.3. Rotation of the waveplates
We consider a small additional angle in the rotation of thelth waveplate, we then haveαl =αl,0 +δαl . After calculations, we find
Iλ = |Tr −Tα |2 ∣
∣Al(
δαl cos2αl −δα2l sin2αl
)∣
∣
2. (31)
10 20 30 40 50 60 70 80 90 1001.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4x 10
6
Number of simulations
Reje
ctio
n r
ati
o
Fig. 8. Rejection ratio with randomly-chosenε1, j andε2, j.
This shows that the rejection ratio will be much less sensitive to waveplate rotations if cos2αl =0, as shown in Fig. 9(a). The required accuracy in waveplate rotation should not be the limitingfactor in an actual set-up.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−3
104
106
108
1010
1012
1014
Rotation angle of the waveplate, δαl
Reje
ctio
n r
ati
o
Beam 1, 2α1 = 7π/6
Beam 3, 2α3 = π/2
1 2 3 4 5 6 7 8 9 10
x 10−4
105
106
107
108
Differential birefringence, δB/λ
Reje
ctio
n r
ati
o
(a) (b)
Fig. 9. (a) Rejection ratio when one of the waveplates is rotated with respectto its normalposition for 2α1 = 7π/6 (solid line) and 2α3 = π/2 (dash-dot line) and (b) rejection ratiowith a differential birefringenceδB.
6.4. Differential birefringence
Suppose that the birefringence of thelth waveplate is slightly different from the birefringenceof the other waveplates,Bl = B0 +δB. The detected intensity is then given by
Iλ =
∣
∣
∣
∣
Al sin2αlπδB
λ
∣
∣
∣
∣
2
. (32)
As shown in Fig. 9(b), we should haveδB/λ ≤ 5×10−4 in order to have a rejection ratiohigher than 106, which should be easier to reach in the infrared than in the visible region of thespectrum.
7. Conclusions
We have derived an expression for the generalizedN-beam nulling condition, which includesamplitude, phase and polarization. We have applied this condition to the cases of a two- and athree-beam nulling interferometer. We have shown that interferometry with beams with differ-ent coherent states of polarization is possible with a theoretically perfect contrast.
We have seen that we can theoretically reach an infinite rejection ratio in the monochromaticcase without any phase shifter, using only polarization rotation. In a wide spectral band, wecan still have a high rejection ratio without any phase shifter, but we need an extra device,which could be an achromatic polarization rotator. The approach we show involves only com-mercial elements: linear polarizers and waveplates. We thus have introduced a totally new typeof nulling interferometers, which should allow a high rejection ratio, without any achromaticdevice. We have derived an expression for the nulling condition of the newly-designed nullinginterferometer, which should be fulfilled only by rotating waveplates.
We have looked at theθ -dependence of the transmission map and have seen that, as inregularnulling interferometers, the only three-telescope configuration to have aθ 4-transmission mapis the linear configuration, which is not very interesting for exoplanet detection. As conclusion,the proposed design does not need any amplitude-matching device and is also very suitable forfast internal modulation. Indeed, it allows to obtain a continuous set of transmission maps, byonly rotating the waveplates. But a modulation strategy still has to be investigated.
Another interesting aspect of these calculations is that there is no theoretical limit for thewidth of the spectral band for which we reach a deep null. However, the transmission is notequal for all wavelengths. We have established a criterion to define the acceptable spectralband in such a way that the transmission over the whole spectral band is higher than half themaximal transmission. We have seen that this criterion could lead to a relatively wide band inthe infrared. The ultimate limit of the spectral band will probably be set by the transmission ofthe linear polarizers and the waveplates.
In order to complete the analysis, we also have looked at the sensitivity of the proposeddesign to some imperfections and misalignments. We have seen that the factor limiting therejection ratio will probably be the imperfections of the polarizers. Indeed, rotations can bereached very accurately. Phase mismatching and differential birefringence can be a problem inthe visible region but are much less important in the infrared. In conclusion, using the approachpresented in this paper, we have shown that it should be possible to reach a high rejectionratio without any achromatic device, thus opening a new promising way to detect Earthlikeexoplanets.
Acknowledgments
This research was supported by theKnowledge center for Aperture Synthesis, a collaborationof TNO and Delft University of Technology, The Netherlands.