Sub-Rayleigh Optical VortexCoronagraphy

E. Mari1, F. Tamburini2, G. A. Swartzlander Jr3, A. Bianchini2, C.Barbieri2, F. Romanato4 and Bo Thide5

1 CISAS - Centro Interdipartimentale di Studi e Attivita Spaziali ”G.Colombo”, University ofPadua, Via Venezia 15 35131, Padova, Italy

2 Department of Astronomy, University of Padua, Vicolo dell’Osservatorio 3, 35122 Padova,Italy

3 Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology, 54 LombMemorial Drive, Rochester, New York 14623, USA

4 Department of Physics, University of Padua, Via F. Marzolo 8, 35131 Padova, Italy5 Swedish Institute of Space Physics, Box 537, Angstrom Laboratory, SE-75121 Uppsala,

Swedenelettra.mari@unipd.it

Abstract: We numerically estimate the theoretical efficiency of an opticalvortex coronagraph with a N-step spiral phase plate in its optical path inthe super-resolution regime, i.e., when the separation of the two sources arebelow the Rayleigh separability criterion. In this condition this coronagraphis expected to be quite efficient. The contrast achieved by the instrumentdecreases from its ideal limit, a fraction of the light from the secondarysource is detected and it increases with the number of steps and the angularseparation of the two stars.

© 2010 Optical Society of AmericaOCIS codes: (050.4865) Optical vortices; (100.6640) Superresolution; (050.1970) Diffractiveoptics; (350.1260) Astronomical optics

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1. Introduction

Optical Vortices (OVs) are phase singularities embedded in light beams with helicoidal wave-front [1]. Beams harboring OVs carry a quantity of orbital angular momentum (OAM) of `h perphoton [2], where ` is the topological charge. The amplitude of the electromagnetic (EM) fieldof an OV can be described as a superposition of Laguerre-Gaussian (L-G) modes characterizedby the two integer numbers ` and p. The azimuthal index, `, indicates the number of twists ofthe helical wavefront within the space of a wavelength and the quantity p the number of radialnodes present in the mode. Written in cylindrical coordinates, the amplitude of the EM field isgiven by

u`p(r,θ) ∝

(r√

2w0

)|`|L|`|p

(2r2

w20

)exp(− r2

w20

)exp(−i`θ), (1)

where w0 is the beam waist, and L|`|p is the associated Laguerre polynomial of degree p. ThePoynting vector of the field is classically sketched as it were precessing around the optical axis[3, 4] along which the beam is propagating.

In the last two decades, the orbital angular momentum of light found interesting applicationsin different research fields such as quantum information [5, 6, 7], free-space communication[8], biology [9] and microscopy [10]. Recently, OVs attracted the attention of the astronomicalcommunity, after the seminal paper by Harwit [11], and with more recent pioneering works in

this subject [12, 13, 14]. Superresolution is one of the most striking astronomical applicationsprovided by the topological properties of OVs. Recently, it was demonstrated that, imposingOAM to the light collected by a telescope, the resolving power of any optical instrument (tele-scope, microscope, etc.) can be improved up to an order of magnitude below the Rayleighcriterion limit with white, incoherent, light and even up to 50 times when monochromatic andcoherent sources are used [15]. Another striking application of the optical vorticity of light isthe application of these doughnut-shaped rings of light intensity distributions for the detectionand direct imaging of extrasolar planets [16, 17, 18, 19]. In this paper we show, through numer-ical simulations, that an optical vortex coronagraph can ideally detect faint sources even muchabove the Rayleigh criterion.

In Sect. 2, we describe the different types of phase mask designs used to produce OVs.In Sect. 3 discuss the optical design of the instrument and study its properties by simulatingthe effects of two equally luminous sources placed at different angular distances in the sub-Rayleigh regime and then draw our conclusions.

2. The Optical Vortex Coronagraph

Coronagraphs are optical instruments designed to suppress the light of a star, set on-axis, toreveal the presence of fainter objects around it, such as the planets harbored by a star [16].In the recent years were developed several coronagraphic concepts characterized by differentmethods of rejection of the stellar light [20, 21, 22, 23]. The optical vortex coronagraph (OVC)exploits the topological properties of OVs to faint the light of an on-axis star and reveal nearbyfainter objects [17, 18]. This setup gave striking proofs of its validity by imaging planets arounda distant star with a vortex vectorial mask [19]. Fig. 1 shows the optical scheme of an OVC.

Fig. 1. Optical scheme of an OVC. T is the telescope. L1 the collimating lens, LS the Lyotstop, placed at the same focal distance as that of L1. The light of the object on-axis, O1,crosses the central dislocation of the SPP and is transformed into a concentric superpositionof optical vortices then blocked by LS. The light emitted by the off-axis source, O2, skipthe central dislocation of the SPP and the LS, so it is focused by L2 to be detected.

For the detection of terrestrial exoplanets it is necessary to reach an attenuation around10−9−10−10 that can be reached by the OVC in ideal conditions [24, 18, 25]. There are differ-ent ways to impose optical vorticity to a light beam. Here we analyze and discuss the propertiesof a particular vortex lens, the spiral phase plate (SPP) [26]. The SPP is a transparent helicoidalplate whose thickness increases as the azimuthal angle θ increases, with the shape of an heli-coidal staircase made with N steps (here and thereafter multi level SPP, MLSPP). Each of thesesteps introduce a constant phase variation in the light. The limiting case is when the numberof steps tends to infinity, and the spiral shape of the SPP has a smooth, continuous, elicoidalpattern. In this case, the phase of the light is modulated smoothly and continuously. In this

case, the ideal continuous spiral phase plate (CSPP) imposes a precise topological charge to thebeam, ` given by

`=∆n ·hs

λ(2)

where hs is the jump of the CSPP, λ the wavelength of the incident light beam and ∆n thedifference between the refractive index of the SPP and that of the surrounding medium.

The discontinuities that build a MLSPP introduce a degree of optical vorticity different fromthat of a CSPP, being the total result a finite superposition of discrete phase shift contributions.For this reason the relationship between the total topological charge, `, and the wavelength ofthe beam, λ , now depends on the number of steps used to build the whole phase gap [27],

`=∆ ·hs

λ

(N +1

N

). (3)

When a monochromatic Gaussian light beam crosses a SPP in its central dislocation, the lightis transformed into a superposition of L-G modes or, when Fraunhofer diffraction patterns areproduced by Gaussian beams, in a superposition of Kummer modes [28]. For any integer valueof the topological charge `, the intensity profile of these vortices present a doughnut-like shape,characterized by a circular symmetry with a central dark zone. When the beam, instead, crossesthe phase mask off-axis, the intensity profile looses its axial symmetry and the resulting OVbecomes a superposition of different L-G modes, generated by the different parts of the beamcrossing the mask in different places. When a beam is crossing the vortex lens far away from itsoptical singularity, it will behave as it were crossing a simple slab of glass with the result thatthe intensity distribution of the beam would tend to preserve more its Gaussian profile [29].

3. Sub-Rayleigh coronagraphy

In this section we show the results of the numerical simulations of an OVC mounted at adiffraction-limited telescope. Two monochromatic sources, placed at different angular sepa-rations, generate two ideal Airy patterns that cross the vortex lens. One source is always kepton-axis and is blocked by LS. The routines, written in IDL, take also in account the numberof steps present in the mask design. The images are encoded and manipulated by using a setof 4096× 4096 matrices. The two Airy patterns are transformed into a superposition of LGmodes, that reproduce the effects of the SPP with a given number of steps, N. The OVs thenare collimated to the Lyot Stop, having a diameter 0.8 times than that of the exit pupil of thetelescope [18]. We see that the more the secondary star is closer to the primary, the less is itslight passing through the LS, making its detection more and more difficult.

With the numerical simulations we have calculated the percentage of the light of the off-axis source passing through the LS by varying the number of steps N and the distance of thesecondary star, expressed in fraction of λ/D, where D is the aperture of the telescope. Tab.1reports the numerical results obtained for the ideal CSPP, (N→ ∞), and four types of MLSPPswith 8, 16, 64 and 512 steps, obtained at different separations of the two sources.

From the calculated percentage of light, we see that the OVC has a good efficiency at angularseparation larger than 2λ/D: at this distance it eliminates the light from the main on-axis starwith a minimum loss of the light of the planet. What is interesting is that a certain percentageof the light from the secondary source passes through the LS to be detected even below theRayleigh criterion. The quantity of light increases with the number of step of the SPPs. Withan ideal CSPP and for a distance of 0.5λ/D, the percentage is around 18%. This would meanthat one could detect an exoplanet with a contrast of 1010−1011 even at very small separationangles, much below the Rayleigh criterion limit, favoring the study of stellar systems similar tothe Solar System, permitting to image even planets in stellar systems at large distances.

N-levels / d 0.5 1 1.22 1.5 2 3∞ 18.04 56.03 69.58 79.58 89.71 99.03

512 18.02 55.97 69.48 77.48 89.63 98.9764 17.83 55.47 68.81 76.81 88.81 98.4316 16.30 51.64 64.11 72.11 82.8 94.948 12.25 41.31 53.82 61.13 72.92 89.99

Table 1. The percentage of the light of the secondary source passing through the Lyot stop.It depends on the number of the steps used to build the total phase gap and on the distanced in units of λ/D.

3.1. Graphic results

Here, we report some of the intensity plots obtained in the numerical simulations of Tab. 1below (d = 0.5) and around the Rayleigh criterion limit (d = 2). .

Figure 2 shows the results obtained with a CSPP in the ideal case when around the primarystar is found a companion with intensity 10−8 times lower that of the main source. From thesimulations we see that MLSPP do not reach enough contrast to image such a faint object atd = 0.5 [24].

Fig. 2. Coronagraphic image (2D and 3D) of a CSPP OVC. In the two left panels thedistance between two objects is below the Rayleigh criterion, 0.5λ/D, in the two rightpanels the distance is 2λ/D. At distance 0.5λ/D is visible just 18% of the light of planet.At 2λ/D around 90% of the detectable light of the secondary source passes through theLyot stop.

Figure 3 and Fig. 4 report the results obtained with two MLSPPs, having 8 and 64 levels, withintensity ratios 1 and 10−4 respectively, to simulate realistic conditions that can be obtained atthe telescope. In this latter situations, when d = 0.5λ/D, the percentages of the light of thesecondary sources passing through the LS are around 12.25% and 17.83%.

Fig. 3. Simulated coronagraphic image in 2D and 3D for a MLSPP with 8 levels, when thetwo sources have the same intensity. Distances between two objects: 0.5λ/D and 2λ/D.

Fig. 4. Coronagraphic image in 2D and 3D for a MLSPP with 64 levels. The two sourceshave an intensity ratio of 10−4.

In any of these plots, we see that at d = 0.5λ/D, the OV produced by the on-axis source ispartially filled by a portion of the light of the fainter companion, revealing its presence. Ourresults suggests that the OVC could be applied to the detection of the unresolved companionsin double systems and to the study of the transit of planets.

Conclusions

In this paper we have numerically studied the efficiency of the OVC simulating the separationof two sources at different distance with different phase masks. In the simulations we studiedthe instrument as working in the monochromatic regime or, equivalently, equipped with anachromatic vortex lens.

We have shown that, in any of the optical configurations considered, always a fraction ofthe light from the fainter companion starts becoming visible even below the Rayleigh criterionlimit. The percentage of light from the secondary star passing through the LS increases withincreasing angular separation. For that reason we suggest that OV coronagraphy could be asa concrete method to detect faint unresolved companions with space telescopes, where thedetrimental effects of the atmospheric turbulence do not affect the profile of the OVs. Thepossibility of observing solar systems below the Rayleigh criterion will open new frontiers inthe study of planet transit.

4. Acknowledgements

This work has been partially supported by the University of Padova, by the Ministry of Univer-sity and Research and by the CARIPARO Foundation inside the 2006 Program of Excellence.