THE OPTICAL DIFFERENTIATION CORONAGRAPH

Jose E. Oti, Vidal F. Canales, and Manuel P. Cagigal

Departamento de Fısica Aplicada, Universidad de Cantabria, Avenida Los Castros S/N, E-39005 Santander, Cantabria, Spain;

otije@unican.es, fernancv@unican.es, perezcm@unican.es

Received 2005 January 31; accepted 2005 May 13

ABSTRACT

We describe a new stellar coronagraph based on the standard coronagraph scheme but transformed to perform theoptical differentiation of the incoming field. It offers a new method to detect exoplanets providing both deepstarlight extinction and high angular resolution. To perform optical differentiation, the coronagraph occulting diskis replaced by a differentiation mask. Although the theoretical rejection rate of our coronagraph is infinite, nu-merical simulations of the perfect case, with no phase errors and perfect telescope pointing, are carried out showingthat on-axis starlight is reduced to very low intensity levels corresponding to a gain of at least 37 mag (10�15 lightintensity reduction). To take full advantage of this design, the distortions of the atmosphere must be reduced eitherby the use of extreme adaptive optics or space telescopes, although the latter solution is preferred. The deep starlightextinction is mandatory to achieve direct detection of Earth-like exoplanets and, what is more important, to performspectroscopy of their atmosphere and look for habitable conditions or signs of life.

Subject headinggs: astrobiology — instrumentation: adaptive optics — instrumentation: high angular resolution —planetary systems — techniques: high angular resolution

1. INTRODUCTION

The list of discovered extrasolar planets orbiting Sun-likestars has increased continuously since the first detection by in-direct methods.1 The final goal of exoplanet detection is to ob-tain direct images of Earth-like planets around solar-type stars inorder to perform spectroscopy of its atmosphere and look forhabitable conditions and signs of life (Wolf & Angel 1998).Detection of such faint objects is a formidable task because ofthe small angular separation and the large contrast ratio betweena Sun-like star and an Earth-like planet. In the mid-infrared, theintensity contrast ratio, of 10�6, is minimal (Beichman et al.1999). At visible wavelengths, i.e., 0.3–1.1 �m, this contrastratio is more striking, 10�9 to 10�10.

Currently projected space missions focused on the direct de-tection of Earth-like planets, such as the Terrestrial Planet Finder(TPF ) NASA program (Beichman et al. 1999), rely either onnulling interferometry or on stellar coronagraphy. Despite therelatively better conditions for exoplanet detection that the mid-infrared presents, visible wavelengths are preferred for a first TPFdue to the advantages that they offer, and as a precursor of an in-frared nulling interferometermission (Kuchner&Spergel 2003a).At visible wavelengths, the required diffraction-limited resolu-tion is attained with a smaller telescope than at infrared wave-lengths. Furthermore, the exozodiacal-dust background noise isrelatively less important at visible than at the mid-infrared, andthe imaging capabilities of conventional optical telescopes makethis noise easier to handle than in a mid-infrared TPF interferom-eter. Moreover, optical detectors require much less thermal con-trol than infrared detectors. On the contrary, a visible-wavelengthcoronagraph requires a level of wave front control higher thanfor a mid-infrared interferometer. Finally, the available biomark-ers O2 and O3 at visible wavelengths are more useful than thebiomarker O3 in the mid-infrared (Des Marais et al. 2001). Thevisible-wavelength TPFs proposed to date can be generally clas-sified as stellar coronagraphs. Stellar coronagraphy, originallydeveloped to study the Sun, involves the use of a coronagraph to

image the surroundings of a star (a good description is presentedin Sivaramakrishnan et al. 2001). A coronagraph consists of atelescopic system plus an imaging lens. It is designed to obstructthe light of the star by means of an occulting disk placed on thestar’s intermediate image. The light diffracted by the occultingdisk spreads on the edge of the pupil and is removed by placinga circular opaque mask, the Lyot stop (Lyot 1939), in the exter-nal part of the relayed pupil. Finally, the remaining light formsthe image of the faint region around the central star that wouldotherwise be occulted by the star halo. Modern coronagraphs usespecially designed Lyot stops that match the residual image ofthe star in the Lyot plane, which depends on the telescope pupiland the Fourier transform of the occulting mask. The Lyot stopradius is an important parameter, since it is necessary to find aradius that is small enough to ensure that the star background isdarker than the faint source but large enough to reduce the planetlight as little as possible. In past years, new coronagraphic con-cepts were described. These proposals use a specially designedfocal-plane phase mask to achieve better starlight backgroundreduction. Roddier & Roddier (1997) proposed a phase maskcovering the inner part of the star image, allowing better back-ground light reduction than the standard coronagraph. Furtherstarlight extinction could be achieved by the Achromatic Interfero-Coronagraph (AIC) (Gay & Rabbia 1996; Baudoz et al. 2000) orby the Four-Quadrant PhaseMask coronagraph (FQ-PM) (Rouanet al. 2000), which uses an alternative phase-step mask arrangedin a quadrant scheme. Kuchner & Traub (2002; see also Kuchner& Spergel 2003b) described another method that uses band-limited masks, and it presents promising results. Theoreticallythose techniques, like the herein presented optical differentia-tion coronagraph, achieve perfect cancelation of the starlight.

Here we present a novel concept that consists of the imple-mentation of an optical differentiation system in a conventionalcoronagraph. This is accomplished by using an appropriate dif-ferentiation mask instead of the standard occulting disk. It pro-vides both high angular resolution and high dynamic range, andtheoretically, it allows perfect starlight extinction. Our instru-ment is expected to permit the direct detection of Earth-like exo-planets and to perform spectroscopy of its atmosphere. The use1 For a complete list, see http://exoplanets.org.

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# 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

of space-based telescopes or extreme adaptive optics (Cagigal& Canales 2001) is mandatory to reduce atmospheric distortionsin order to take advantage of the capabilities of our coronagraph.Simulations under ideal conditions are carried out and show thatour instrument performs at least as well as other coronagraphs.In x 2 the principle of operation of the coronagraph is presentedand a complete description of the differentiation mask is shown.Then numerical simulations are carried out, and their results onparent-star suppression and, as an application example, success-ful exoplanet detection are presented in x 3. In x 4 we examinethe exoplanet detection procedure and the various sources oferror. Last, in x 5 some conclusions are derived.

2. THEORETICAL DESCRIPTION

Our instrument is based on a standard coronagraph inwhich theocculting mask is replaced by a linearly increasing amplitude-transmittance mask. From the definition of the Fourier transform(FT), the derivative of a one-dimensional function f (x) can beexpressed as

df (x)

dx¼

Z 1

�12�uxiF(ux)e

i2�uxx dux: ð1Þ

F(u) represents the Fourier transform of f (x), ux is the spatialfrequency coordinate in the x-direction, and i is the imaginary unit.This expression can be easily extended to the two-dimensionalcase. Equation (1) implies that a mask, whose amplitude trans-mittance linearly increases along one direction, placed at thecommon focal plane of a telescopic system performs the opticaldifferentiation of the incoming field along the direction of themask slope (Iizuka 1987; Oti et al. 2003). The novel idea in ourproposal is to transform a coronagraph into a system able toperform optical differentiation, thereby improving its capacity forexoplanet detection. Indeed, the wave front phase of an unre-solved star on the optical axis of this system is completely planeover the entrance pupil, and hence its derivative is zero. On theother hand, the wave front phase of an exoplanet is tilted withrespect to that of its parent star, and as a consequence, its deriv-ative is different from zero. Hence, we can take advantage of thisfact to develop an easy and usefulmethod to reveal the presence ofexoplanets.

Let us follow the path of the electric field as it passes throughthe coronagraph depicted in Figure 1. The incoming field f (x, y)(plane a) at the entrance pupil of the coronagraph is Fourier-transformed by the first lens. Then the transformed field F(ux,uy) (plane b) is multiplied by a mask of the form 2�ux i placed atplane c. This product is Fourier-transformed again by the sec-ond lens. Equation (1) provides the derivative of the incomingfield along the direction of the mask slope, df (x, y)/dx, at theLyot stop plane (plane d ). The Lyot stop shrinks the pupil toblock the light that is diffracted by the mask. Finally, the thirdlens on the coronagraph is used to form the image onto thedetecting system such as a CCD camera in plane e.

The field at the entrance pupil is E(x; y) ¼ E0ei�(x; y), where

�(x, y) is the wave front phase and E0 is the constant ampli-tude. The detected intensity at coronagraphic pupil (plane d inFig. 1), I(x, y), is closely related to the wave front phase de-rivative of the incoming field. From equation (1) it follows that

I (x; y) ¼ FT 2�uxiFT E(x; y)½ �f gj j2

¼ @E(x; y)

@x

��������2

¼ E0j j2 @�(x; y)

@x

��������2

; ð2Þ

where FT stands for the Fourier transform operation. The coro-nagraph performs a perfect cancelation if the phase �(x, y) isconstant so its derivative is also equal to zero. However, a tiltedsource presents a phase different than zero, which provides a de-rivative estimate different than zero. Therefore, the light comingfrom the centered star is removed, allowing us to search for apotential faint object in the circumstellar environment.

2.1. Mask Description

At this point, a further description of the differentiation maskshould be made. The mask used to perform optical differentia-tion is a complex mask whose amplitude linearly increases alongthe direction where the derivative is to be performed as describedin equations (1) and (2). It can be expressed as 2�ir/(kf ), where rrepresents a real distance in the mask plane, k is the wavelengthof the incoming light, and f is the focal length of the trans-forming lens. The actual differentiation mask ranges from pos-itive to negative values so it must be realized using two differentcomponents: an amplitude transmittance mask proportional toj2�r/(kf )j and a phase mask consisting of a phase step of ��/2for the negative valued coordinates and of +�/2 for the positiveones. This phase step provides the imaginary unit required toattain the correct sign of the differentiation mask. The phase stepresembles that used in the achromatic phase knife coronagraph(Abe et al. 2001). Since the actual transmittance mask rangesfrom 0 to 1, the mask slope is determined by the size of the mask.Figure 2 shows the different components of the differentiationmask and its final shape. It is important to note that the differ-entiation mask has a finite size, and as a consequence, it presentssharp edges that diffract the incoming light. Although computersimulations show that this mask configuration allows star ex-tinctions up to 108, it is hard to recover the exoplanet from thestarlight background, because the planet itself suffers absorp-tion by the mask. In order to overcome this limitation we proposeto multiply the optical differentiation mask by a Gaussian pro-file as shown in Figure 2. This Gaussian profile varies only alongthe differentiation direction. It smooths the differentiation maskborders, and consequently, it reduces the amount of diffractedlight, which allows star extinctions rates of 1015 for the maskshown in Figure 2. The Gaussian profile is multiplied by a scal-ing factor in order to get an actual mask ranging from �1 to 1.Another effect of the latter mask is to reduce absorption of theexoplanet light by increasing the mask slope at its central region.Furthermore, the Gaussian profile concentrates the diffractedlight on a narrower area near the coronagraphic pupil borderso it can be removed more efficiently by the Lyot stop. SmallGaussian widths produce a spread of the diffracted light, andconversely, large Gaussian widths narrowly concentrate it. Ex-tinction rates may vary largely depending on the width of theGaussian profile and the size of the mask. In general, the largerthe differentiation mask size, the deeper the extinction rateachieved. The Gaussian width is selected so the borders of themask end smoothly avoiding sharp mask edges. Another effect

Fig. 1.—Set-up of the proposed stellar coronagraph. It is composed of threelenses (L1, L2, and L3), the Lyot stop (LS), the differentiation mask (DM), placed atthe common focal plane of L1 and L2, and an imaging system like a CCD camera.

OTI, CANALES, & CAGIGAL632 Vol. 630

of the use of the Gaussian is to reduce the field of view atthe outer edges (about 35k/D for the mask plotted in Fig. 2). Thelight of the centred star is blocked by the opaque region of themask but the light coming from other sources is only partiallyabsorbed, depending on their positions. The remaining on-axislight is bounced out of the central region of the pupil (Fig. 3b)that is blocked by the Lyot stop (Fig. 3c), producing deep star-light extinction on the final image (Fig. 3d ).

3. NUMERICAL SIMULATIONS

We check the performance of our coronagraph by means ofnumerical simulations using the fast Fourier transform (FFT)

Fig. 3.—Simulated images at different planes in the optical differentiationcoronagraph illustrating its principle of operation. (a) Image of the star PSFmultiplied by the modified differentiation mask. (b) Intensity distribution justbefore (b) and after (c) the Lyot stop plane. (d ) Final image detected at theCCD plane. Images are displayed in different intensity scales.

Fig. 2.—Design of the differentiation mask required in our instrument.(a) Amplitude transmittance of the differentiationmask (solid curve) and the sametransmittance mask multiplied by a Gaussian profile (dashed curve). (b) Phasejump needed to obtain the imaginary unit. (c) Differentiation masks obtainedplacing together the masks of (a) and (b). The results are the standard differ-entiation mask (solid curve) and the modified differentiation mask (dashedcurve) that allows starlight extinction deep enough to uncover exoplanets.

Fig. 4.—Radial intensity profiles at 45� with respect to the x-axis. Intensityprofile obtained with the optical differentiation coronagraph (solid curve) inthe perfect wave front case. For comparison, the dashed curve shows the Airyprofile, taking into consideration the Lyot stop of the coronagraph.

OPTICAL DIFFERENTIATION CORONAGRAPH 633No. 1, 2005

routine (Press et al. 1995). Arrays of 1024 ; 1024 data pointswere used to avoid aliasing effects and to achieve good sam-pling of the image plane. The coronagraph entrance pupil issimulated with a data sampling of 128 ; 128. First, the FFT ofthe field at the entrance pupil of the system is carried out, andthen it is multiplied by the differentiation mask (Fig. 3a). Nextanother FFT is performed and the resulting pupil is apodizedwith the Lyot stop (Fig. 3c). Last, to obtain the final corona-graphic image, an additional FFT is made and the intensity ofthe field is calculated (Fig. 3d ). The value of the Lyot diameteris an important factor to be considered. A large Lyot stop radiuswill allow too much starlight to pass trough the system, and as aconsequence, its performance will be degraded. On the otherhand, if the Lyot radius stop is too small, much of the light fromthe planet will be blocked, making its detection even moredifficult. Moreover, this decrease in the detected exoplanet lightis not accompanied by great improvement of starlight reduc-tion. We have found that an appropriate radius of the Lyot stoplies around the 85% of the entrance pupil radius. The Lyot stopcould be optimized to match the shape of the image at the pupil(Fig. 3b), thus improving the exoplanet light throughput with-out losses in starlight extinction. The optimum Lyot shape re-sembles that described by Kuchner & Traub (2002).

In Figure 4 we present the results of numerical simulations ofour instrument in the perfect wave front case. Under theseconditions our coronagraph allows deep starlight backgroundreduction. This implies an easier detection of faint objects in thestellar surroundings. The differentiation mask used in thesesimulations, shown in Figure 2, is the standard differentiationmask multiplied by the Gaussian profile. It has a radius in thex-direction of about 50k /D and the Gaussian profile has a var-iance of �2 ¼ 100(k /D)2. Next, as an application example, we

apply our technique to directly detect an Earth in a system likethe solar system at a distance of 60 pc working at a wavelengthof 0.5 �m in a 10m ground telescope or, equivalently 38 pc for a6m space telescope under ideal conditions (no wave front phaseerror and perfect optics). The simulated contrast ratio betweenthe exoplanet and its parent star is of 22.5 mag (10�9 intensityratio). A deep extinction of 37 mag (10�15 of intensity reduc-tion) is achieved and the faint exoplanet is clearly visible, asillustrated on Figure 5.

4. ANALYSIS

For our coronagraph, the angular distance where the off-axiscompanion transmission is decreased by a factor of 2 (inner work-ing angle) is about 4.8k/D, which is similar to the band-limitedcoronagraph (Kuchner & Traub 2002). Nevertheless, despite theabsorption produced by the differentiation masks, our instrumentis able to successfully detect very faint sources as near the parentstar as 0.01k/D (with a planet flux reduction of about 6 orders ofmagnitude) providing high angular resolution. This implies thatexoplanets could be detected in even more distant stars, thus in-creasing the number of surveyed stars. According to equation (2),since our instrument estimates the wave front derivative, the in-tensity detected from the planet decreases when its wave fronttilt decreases. Let us define a line from the star to the planet. Theintensity detected from the planet is a maximum when the star-to-planet line is parallel to the mask slope direction. As the maskis rotated, the projection of this line over the mask’s slope direc-tion decreases, and consequently the planet signal fades out untilit completely disappears at a rotation angle of �/2. This effect ischecked using a computer simulation, and its results are shownin Figure 6 for a rotating planet (which is equivalent to a rotatingmask). The top panel in Figure 6 shows the normalized peakintensity for a rotating exoplanet. The distance between star andplanet remains constant and is about 1.5k /D in the simulation. Inaccordance with theory, the maximum detected intensity corre-sponds to a star-to-planet line parallel to the mask slope direction(Fig. 6a). As the planet rotates, the detected intensity decreases(Figs. 6b and 6c). For a rotation angle of �/2 the signal from theplanet disappears (Fig. 6d ). This behavior can be easily ex-plained in terms of the squared modulus of the star-to-planet lineprojection over the mask slope direction. It can be seen that thecurve in Figure 6 remains unchanged for different exoplanet an-gular separations. Furthermore, the greater the angular separa-tion between star and exoplanet, the larger the detected exoplanetpeak intensity. A mask rotation from 0 to �/2 provides a com-plete survey of all possible exoplanet locations, minimizing thenumber of images required for a successful discovery. Hence, theprocedure to uncover all the existing exoplanets orbiting the staris to use a rotating mask. Moreover, the recorded images of theexoplanet give information on key parameters of its orbit. To per-form spectroscopy most efficiently, the star-to-planet line shouldbe parallel to the slope mask direction so the planet detected in-tensity is a maximum.

4.1. Sources of Error

As in other proposed coronagraphs (Roddier & Roddier1997; Rouan et al. 2000; Kuchner & Spergel 2003b), pointingerrors are critical, since small displacements of the star from thecenter imply a growing light background that overwhelms thefaint exoplanet. To confirm this idea we have performed com-puter simulations including pointing error. We have displacedthe star out of the optical axis of the coronagraph, and then wehave estimated the intensity of the coronagraphic image at the

Fig. 5.—Computer simulation showing the exoplanet detection capability ofthe novel optical differentiation coronagraph. The star/exoplanet contrast ratiois 109, and its angular separation is 1.5k /D. (a) When no coronagraphic systemis present the star background completely overwhelms the exoplanet signal.(b) Our coronagraph achieves starlight extinction rates of �1015 (�37 maggain), and consequently the exoplanet undoubtedly appears. The intensityprofiles in the x-direction of the star (solid line) and the exoplanet (dashed line)are represented in the bottom section of the figure. The white cross marks theposition of the parent star. The intensity scales of the images are different inorder to stand out the planet image. The modified differentiation mask used inthis simulation is the one shown in Fig. 2. The simulations were carried out for aperfect wave front.

OTI, CANALES, & CAGIGAL634 Vol. 630

star position. A small pointing error significantly degrades theperformance of the optical differentiation coronagraph. For in-stance, a pointing error of 1:2 ; 10�3k /D increases starlight in-tensity to 10�8, as shown in Figure 7 for the perfect wave frontphase case. The displacement of the star is performed on theslope mask direction. Perpendicular displacements do not de-grade the performance of our coronagraph because it is notsensitive to those pointing errors. Starting from our differenti-ation concept it is possible to define a mask 2�r 2/(kf )2 that oncemultiplied by a Gaussian profile is more robust to pointing er-rors than our mask. This mask resembles those proposed byKuchner & Spergel (2003b). However, the latter increases thelight absorption near the parent star, and consequently the innerangle is larger than the herein proposed mask. A procedure toincrease the accuracy of the telescope pointing could be imple-mented monitoring the final detected starlight intensity at the

coronagraph CCD. Using the standard pointing procedure, aminimum of the starlight intensity must be obtained. Once theinstrument has been successfully pointed, a rotation of the dif-ferentiation mask must not vary the on-axis star intensity level.In spite of the pointing error, the differentiation coronagraphstill provides the derivative of the incoming field, thus allowingthe discovery of exoplanets, provided that the starlight intensitylevel is maintained below the exoplanet intensity.

The differentiation mask must be fabricated accurately, es-pecially the opaque region of the amplitude mask, due to thelarge on-axis light intensity in this area. Errors in the opacityof the mask of just 10�6 will produce a leak of on-axis lightthrough the Lyot stop large enough to cover the planet at a dis-tance of few diffraction widths. Like many of the previouslyproposed coronagraphs (Roddier & Roddier 1997; Rouan et al.2000), the one described in this work is not polychromatic dueto the required �/2 phase step. However, all the solutions to thisdrawback previously developed to be applied in other corona-graphs could be easily implemented in our design. Additionally,the shape of the phase jump does not depend on the wavelength,thus representing a further advantage.

Amore detailed study considering different error sources willbe addressed in a subsequent paper.

5. CONCLUSION

We described a novel stellar coronagraph that consists in amodification of the standard coronagraph scheme in which theocculting disk is replaced by a differentiation mask. This opticaldifferentiation capability provides deep on-axis starlight ex-tinction. The required mask has an amplitude transmittance thatlinearly increases along the direction in which the derivative isto be performed. Itmust be realized using two components: an am-plitude transmittance mask and a �/2 phase step to achieve thecorrect sign of the differentiation mask. Furthermore, multiply-ing the mask by a Gaussian profile improves the extinction rateby decreasing the amount of diffracted light. Numerical simu-lations prove that star extinctions of 37 mag (10�15 intensityreduction) are achieved for a perfect instrument with no phasedistortion. Besides, our novel concept achieves high angular res-olution, which allows the detection of faint sources as near to theparent star as a few times 0.01k/D. This coronagraph is espe-cially well suited for a space telescope experiment. Ground-based

Fig. 7.—Simulated detected intensity with our coronagraph for the perfectwave front case as a function of the star displacement out of the optical axis.The displacement is carried out in the mask slope direction.

Fig. 6.—Simulated images of a rotating exoplanet in the perfect wave frontcase. The star/exoplanet contrast ratio is 109, and the angular separation is1.5k /D. The rotation angles of the exoplanet are (a) 0�, (b) 30�, (c) 60�, and(d ) 90

�. It is important to note that a rotating planet is equivalent to a mask

rotation. The position of the nulled parent star is marked by a white cross.

OPTICAL DIFFERENTIATION CORONAGRAPH 635No. 1, 2005

operation could be also possible with the use of extreme adaptiveoptics systems to compensate for atmospheric distortions. Be-sides atmospheric distortions, pointing error limits the perfor-mance of the coronagraph. However, a procedure to achieve finepointing using the coronagraph’s own signal has been describedhere. The removal of the starlight background to very low levels,under ideal conditions, encourages us to expect that our instru-ment will open the door to search for biomarkers that couldprovide information about the exoplanet habitability or even to

show the existence of life. In conclusion, we believe this in-strument is suitable to detect and perform spectroscopy of theelusive Earth-like exoplanets.

The authors thanks N. Devaney for his valuable commentsand help. This work was supported by Ministerio de Ciencia yTecnologıa AYA2004-07773-C02-01. J. E. Oti was supportedby an FPU grant of the Ministerio de Educacion y Ciencia.

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