Common-path lateral-shearingnulling interferometry with a

Savart plate for exoplanet detectionNaoshi Murakami* and Naoshi Baba

Division of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan*Corresponding author: nmurakami@eng.hokudai.ac.jp

Received June 24, 2010; accepted July 9, 2010;posted August 13, 2010 (Doc. ID 130634); published August 31, 2010

We propose a common-path lateral-shearing nulling interferometer for direct detection of exoplanets. A Savart plateis placed between crossed polarizers to produce a lateral shear and realize fully achromatic and highly stable nullinginterference for starlight. We construct a double-shearing interferometer using two Savart plates for implementingorthogonal x and y shears. A laboratory demonstration is carried out using a broadband light source with a band-width ofΔλ=λ0 ¼ 0:33 (Δλ ¼ 0:2 μm and λ0 ¼ 0:6 μm). As a result, achieved extinction levels are 4 × 10−4 at peak and4 × 10−7 at 10λ0=DL (DL is the diameter of a Lyot stop). © 2010 Optical Society of AmericaOCIS codes: 120.3180, 350.1260, 260.5430, 260.1440.

Exoplanets, orbiting around stars other than the Sun, areinteresting astronomical objects. Direct imaging of exo-planets, especially Earthlike planets, is very challenging,because the exoplanet is much fainter than its parentstar. For direct detection of exoplanets, the bright star-light must be strongly suppressed to reveal a planetarysignal. Thus a dedicated observational instrument calleda high-contrast imager is indispensable. Required con-trast for detecting Earthlike planets would be 1010 inthe visible and near IR.Ground-based extremely large telescopes, such as the

Thirty Meter Telescope and the European ExtremelyLarge Telescope, will be key for imaging Earthlike pla-nets because of their large light collecting power andhigh angular resolution [1,2]. For these future-plannedtelescopes, hexagonal segmented mirrors will be usedfor constructing extremely large primary mirrors. How-ever, starlight suppression with the segmented-mirrortelescope will be difficult because of gaps betweensegments and the shade of a secondary mirror and itssupport structure.Among high-contrast imagers proposed to date, a

lateral-shearing nulling interferometer will be usefulfor exoplanet detection with the segmented-mirror tele-scope, because it does not suffer from complicated pupilfunction [3]. The lateral-shearing nulling interferometerdivides a light beam from a telescope into two beams,and recombines them with a lateral shear and a π phasedifference between them. Thus, an on-axis star can becanceled out, while an off-axis planet survives by virtueof the lateral shear. We note that the lateral shear must bean integer number of the segmented mirror to superim-pose two telescope beams perfectly. For constructing thelateral-shearing nulling interferometer, a modified Mach–Zehnder interferometer has been proposed, in which anachromatic π phase difference is carried out by usingdispersive plates, while a lateral shear is conducted bymoving a mirror in the interferometer [3].We propose to use a Savart plate placed between two

crossed polarizers for constructing the lateral-shearingnulling interferometer. A lateral shear is provided withthe Savart plate [4]. A fully achromatic π phase difference

between the two beams can be realized owing to aproperty of polarization interferometry. Polarization in-terferometry has been applied for focal-plane phase-mask coronagraphs [5,6] and a nulling-interferometriccoronagraph [7].

Figure 1 shows the principle of the proposed method.A common-path optical configuration of the proposedmethod realizes highly stable nulling interference be-tween the two light beams sheared by the Savart plate.A Jones vector of light, after passing through polarizerP1 (θ ¼ 0°) is written as Ei ¼ ½ 1 0 �† († means trans-posed matrix). Note that a 50% energy loss occurs at po-larizer P1 if a planetary signal is unpolarized. The outputlight from polarizer P2 (θ ¼ 90°) is a sum of theoe and eo rays of the Savart plate (here, o and e denoteordinary and extraordinary rays in the Savart plate,respectively). Thus the output Jones vector is given by

Eo ¼ P2ðSoeeiΦx þ SeoÞEi; ð1Þ

where P2 and S are Jones matrices of polarizer P2 and theSavart plate for the oe and eo rays. The term Φx ¼2πsxθx=λ is the phase difference between the two lightbeams at a wavelength λ due to the lateral shear sx

Fig. 1. (Color online) Principle of a lateral-shearing nullinginterferometer based on a Savart plate. States of polarizationat each optical component are also shown.

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0146-9592/10/183003-03$15.00/0 © 2010 Optical Society of America

and the on-sky direction θx. The Savart plate acts as�45°polarizers for the oe and eo rays. Thus, the resultant out-put vector is

Eo ¼12

�0

1 − eiΦx

�: ð2Þ

For an on-axis direction θx ¼ 0, the phase differenceΦx becomes zero for any values of λ. The output Jonesvector then becomes a null vector (Eo ¼ 0), which sug-gests that a fully achromatic and perfect suppressionoccurs for an on-axis pointlike star.For an off-axis planet in a direction of θx ¼ λ=2sx,

where the phase difference becomes Φx ¼ π, construc-tive interference occurs and the output vector becomesEo ¼ ½ 0 1 �†. Thus, the planetary signal can be detectedwithout energy loss, except for the loss due to the polar-izer P1. We note that the constructive interference occursonly for the direction θx ¼ λ=2sx, and the interferometricoutput changes according to sin2ðπsxθx=λÞ. Assumingsx ¼ 1:5 m and λ ¼ 1:6 μm, for a numerical example, con-structive interference occurs for a planet with an angularseparation of 0:11 arcsec from its parent star, which iscomparable to the Sun–Earth system as seen from10 pc away (0:1 arcsec).We carried out a laboratory demonstration of the pro-

posed method. Figure 2 shows an experimental setup.We built a double-shearing interferometer using twoSavart plates in tandem to produce both x and y shears.The double-shearing nulling interferometer is more toler-ant of an apparent stellar size than the single-shearinginterferometer. We note that the energy loss of 50%occurs only at the first polarizer, even for the double-shearing interferometer, despite the overuse of the po-larizers. The optical throughput can be improved byreplacing the first polarizer with a polarizing-beam split-ter to construct a dual-channel configuration.As a model star, we use a 25 μm pinhole illuminated by

a broadband halogen lamp. The light source has a spec-tral bandwidth of about Δλ ¼ 0:2 μm (FWHM) with acentral wavelength of about λ0 ¼ 0:6 μm (i.e., Δλ=λ0 ¼0:33). Behind an entrance pupil with a diameter ofD ¼ 5 mm, the double-shearing interferometer is placedto produce x and y shears of sx ¼ 0:65 mm and sy ¼2 mm. The entrance pupil is reimaged on a plane wherea diaphragm, called a Lyot stop, is placed to extract lightfrom a superimposed area of the pupil. The Lyot-stopdiameter is set to 1 mm, which corresponds to DL ¼2 mm on the entrance-pupil plane (i.e., DL=D ¼ 0:4).

The angular size of the model star is partially resolvedand corresponds to 0:17λ0=DL. Behind the Lyot stop,two exchangeable lenses are used for imaging theLyot-stop plane and the model star onto a CCD camera,respectively.

Figure 3 shows the results of the laboratory demon-stration. Figure 3(a) shows the pupil geometry of the ex-periment seen at the entrance-pupil plane. For the x–yshearing nulling interferometer, the on-sky transmittancecan be written as

Tðθx; θyÞ ¼ sin2�πsxθx

λ

�sin2

�πsyθyλ

�: ð3Þ

A calculated on-sky transmission map assuming a top-hatspectrum from 0.5 to 0:7 μm is shown in Fig. 3(b).

Figure 3(c) shows an acquired destructive (nulled)Lyot-stop-plane image when the model star is placed ontoa central null of the transmission map ðθx; θyÞ ¼ ð0; 0Þ.Achromatic nulling interference occurs over a super-imposed area of the four beams. Figure 3(d) shows aconstructive (bright) Lyot-stop-plane image. The brightimage is acquired by shifting the model star from thecentral null position to ðθx; θyÞ ¼ ðλ0=2sx; λ0=2syÞ.Figures 3(e) and 3(f) show the nulled and bright model-star images after passing through the Lyot stop. A crossin the bright image marks a position of a central null,which suggests that the model star is shifted byðλ0=2sx; λ0=2syÞ. Two white arrows in the nulled imageFig. 2. Experimental setup for laboratory demonstration.

Fig. 3. (Color online) (a) Pupil geometry of experiments; (b)calculated transmission map Tðθx; θyÞ assuming optical para-meters of the experiments; (c), (d) acquired Lyot-stop imageswhen a model star is placed onto a central null and a brightposition of the transmission map; (e), (f) acquired nulled andbright images of a model star.

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indicate ghosts, which are probably generated by theSavart plates or the Glan–Thompson-prism polarizers.Figure 4 shows radial profiles of the acquired bright

and nulled images of the model star as a function ofthe angular distance from the center of each image inunits of λ0=DL. The achieved extinction levels are 4 × 10−4

at peak and 4 × 10−7 at 10λ0=DL. Note that the detector-noise limit lies at a level of about 10−7, as shown by thedashed line.We estimate a theoretical limit of extinction levels by

taking into account the parameters of the experiment, sx,sy, λ0, Δλ, and so on. The estimation suggests that a highextinction level, 2 × 10−5 at peak and 8 × 10−10 at 10λ0=DL,can be realized even for a very large model star with anangular size of 0:17λ0=DL. However, the experimental re-sult is about 1 order of magnitude worse than the theo-retical limit at peak. As seen in Fig. 3(e), the residual lightof the model star seems to be speckled, which suggeststhat phase aberrations of the optical components domi-nantly limit the nulling performance. It is expected thatthe extinction might be improved by an adaptive opticssystem to correct the phase aberrations.For the segmented-mirror telescopes, it is necessary to

design the Savart plate to produce lateral shear with aninteger number of the segmented mirror. However, theshear s is a function of ordinary and extraordinary in-dices of the material and depends on a wavelength ass ¼ sðλÞ. Nevertheless, an error of the lateral sheardue to the dispersion effect is very small and is estimatedto be jsðλsÞ − sðλlÞj=sðλ0Þ≈ 0:032, where λs, λl, and λ0 arethe shortest, longest, and central wavelengths for an ob-servational band. Here, we assume a wavelength rangefrom 1.4 to 1:8 μm and Savart plates made of calcite.

Then, the spilled light due to the dispersion effect canbe blocked by using a Lyot stop with segment gapsslightly wider than those of an original telescope pupil.

The transmission map [Fig. 3(b)] can be regarded as anon-sky variation of the interferometric signal induced bythe path difference between the oe and eo rays of the Sa-vart plate. The transmission map is written by Eq. (3) forsmall off-axis angles of θx and θy. However, it has beenpointed out that the interference pattern will be distortedfor large angles, because nonlinear terms of the off-axisangle will appear in a path difference between the twobeams [8,9]. We estimate the effect of the nonlinear termson the transmission map, and we find that the effect willbe negligible for an astronomical application with verysmall off-axis angles.

We have proposed a lateral-shearing interferometerbased on Savart plates to realize fully achromatic andhighly stable nulling interference. It is expected that theproposed interferometer can be a promising method forboth space- and ground-based exoplanet-finding mis-sions and especially for future-planned extremely largesegmented-mirror telescopes.

We thank K. Oka of Hokkaido University for his manyuseful comments and helpful support. This research wassupported by Hokkaido University and a Grant-in-Aidfor Scientific Research (B) from the Japan Society forthe Promotion of Science (JSPS) (21340041) and JSPSFellows (20·4783).

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Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007),pp. 122–184.

5. N. Baba, N. Murakami, T. Ishigaki, and N. Hashimoto, Opt.Lett. 27, 1373 (2002).

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8. M. Françon, in Optical Interferometry (Academic, 1966),pp. 137–161.

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Applications in Microscopy and Macroscopy (Wiley-Interscience, 1971), pp. 19–34.

Fig. 4. Radial profiles of acquired bright and nulled images of amodel star.

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