The four-quadrant phase mask (FQPM) is an exciting new approach to coronagraphy, a classic astronomical technique for detecting faint companions very close to bright stars. Starlight rejection is potentially very high, and inner working distances are substantially smaller than those achieved with classical Lyot coronagraphy. The key component of the original FQPM scheme is a transparent mask divided into quadrants delivering relative phase shifts alternately 0/π/0/π, inserted in an intermediate focal plane of the telescope. Monochromatic masks of this kind have been successfully demonstrated in laboratory and telescope tests. Fabrication of masks with achromatic π phase shifts is challenging but of great interest for optimum astronomical sensitivity. In this paper I present a novel concept for achromatic FQPM operation that utilizes intrinsic phase relationships between transmitted and reflected beams in a dielectric beam splitter.
© 2005 Optical Society of America
One of the most significant recent developments in astronomy has been the discovery of “extrasolar” planets, now numbering well over 100, orbiting stars other than the sun ; their surprising abundance has inspired planned searches for extraterrestrial life. Developing the instrumentation needed for the eventual direct imaging of planets is a very active field of research, and the requirements are stringent. One of the promising technologies is coronagraphy, for which a number of improvements are being devised and proven, at relatively modest initial performance levels, in current laboratory and telescope testbeds.
2. Lyot and four-quadrant phase mask (FQPM) coronagraphs
Classical (Lyot) coronagraphy, as diagrammed in Fig. 1, may be classified as a form of “amplitude” coronagraphy, placing an opaque spot on the image of the star in an intermediate focal plane. Fourier-optical analysis of the progression of light through the instrument shows that much of the on-axis light is diverted to the periphery of a reimaged pupil, where it can be blocked by an opaque Lyot stop . Images in the final focal plane have the bright central star deeply suppressed, but off-axis sources (companions) are little affected by the coronagraph. An important technical point, generic to all coronagraphs, is that the Airy ring structure of the response pattern from the central star is deeply suppressed as well, effectively attenuating some kinds of speckles .
The Lyot coronagraph can only search for companions further from the star than several times λ/D, so alternatives have been sought. Recently, a very elegant approach using “phase” coronagraphy has been devised: the four-quadrant phase mask (FQPM) . The mask inserted in the intermediate focal plane is now transparent, but delivers phase shifts of 0/π/0/π in alternating quadrants. Such phase shifts may be produced, at least monochromatically, by differential thicknesses of dielectric deposited by thermal evaporation. The operation of the FQPM coronagraph is also described by Fig. 1; it is identical in general configuration to the classical Lyot coronagraph. For an unobscured circular aperture, ideal FQPM mask geometry, and perfect imaging, on-axis light is diverted completely outside the pupil . Tolerances on the FQPM design have been derived [6,7], capturing dependences on wavefront quality, image centering (or residual tip/tilt), fabrication errors in the mask, and other effects. Phase shifts in the focal-plane mask must be very close to π, and requirements on wavefront and residual tip-tilt in the beam feeding the coronagraph make correction of the input beam with an adaptive optics system essential for ground-based FQPM coronagraphy.
Outstanding performance has been achieved by the FQPM coronagraph with monochromatic masks in laboratory tests , inspiring trials at large telescopes [8,9]. At levels of correction available from current adaptive optics systems, monochromatic masks may be used effectively over the fairly substantial bandwidths characterizing standard astronomical interference filters. First observations at Palomar  in the Brackett γ filter (λ=2.168 μm; Δλ/λ~0.01), fed by the high-order PALAO adaptive optics system [10,11], used novel reimaging pre-optics to capture a portion of clear telescope aperture between the envelopes defined by the secondary and primary mirrors, avoiding the obscuration problem and delivering the very high Strehl ratio needed for best FQPM operation. Peak PSF attenuation of order 100 was achieved on the sky, limited by ghost reflections in the focal-plane mask; in the lab, with the mask tipped with respect to the optical beam to eliminate ghosts, attenuation ratios of a few thousand were seen. The monochromatic mask was made by depositing dielectric of index n in quadrants II and IV with extra thickness Δt giving phase
that is, the OPL through the extra dielectric layer is a half-wave greater than through an equal thickness of air. The monochromatic nature of this approach is evident from the form of Eq. (1). With broadband light of spectral width Δλ, the overall starlight rejection factor Rs of such a monochromatic mask may be expected to be limited to 
A number of approaches to achromatic FQPM mask fabrication are being considered, including multi-layer thin-film designs  and use of the “form” birefringence properties of zero-order gratings (ZOGS) . The challenge is not only to deliver a broadband phase difference of π between alternate quadrants of the FQPM mask, but to maintain uniform intensity transmission over the surface of the mask as well. Another approach would be to emulate the operation of dielectric phase plates familiar from their use as achromatic phase shifters in nulling interferometers [13,14]; this appears theoretically feasible with common dielectric materials, but requires layer thicknesses far beyond the normal range conveniently deposited by familiar microfabrication techniques. A novel approach that may be easier to realize in hardware is presented in the next section.
3. Achromatization of the FQPM focal-plane mask: beam splitter phase relationship
Some fundamental achromatic relationships connecting phases of transmitted and reflected beams in a beam splitter follow from symmetries of these devices [15,16]. These relationships have recently been exploited in designs for a broadband phase contrast wavefront sensor  and for a ‘self-nulling’ beam combiner . The latter device would simplify the architecture of planet-finding nulling interferometers, realizing the function of the usual inverting pre-optics through the normal operation of the beam splitters themselves. In the current context, as with the ‘self-nulling’ beam combiner, a very general symmetry (time-reversal) is exploited, so the beam splitters need not have particularly stringent properties to achieve the desired phase performance (unlike the wavefront sensor, for example, which requires a front-back symmetric beam splitter). For intensity balance, a major practical consideration with any FQPM mask, the approach to be presented here will require good matching of R and T, beam splitter intensity (power) reflection and transmission coefficients.
Consider a beam splitter with complex electric-field reflection/transmission coefficients r,r’,t,t’ (i.e., R=|r|2 is the power reflection coefficient, a real number). Primed quantities denote a beam incident on the back surface rather than the front. It is commonly known that t=t’, by the so-called front and back incidence theorem. No such simple relation holds for r,r’. But, for a beam splitter with dielectric (non-absorbing) layers, consideration of time-reversal symmetry indicates that transmission and reflection coefficients rigorously obey 
where the asterisk denotes complex conjugation. As shown in , rr’ and tt for such beam splitters are always opposite in sign, i.e., shifted in phase by π, which is precisely the relative phase required by the FQPM focal-plane mask. The π relative phase relationship holds for any wavelength and for either polarization. Note again that this phase relationship does not place any special requirements on the beam splitter, such as front/back symmetry (which is required by the beam splitter used in a broadband phase contrast wavefront sensor to produce an achromatic π/2 phase shift ).
An implementation of this phase relationship equivalent to an FQPM focal-plane mask is shown conceptually in Fig. 2. Two matched beam splitters, made together in a single fabrication run, generate the beam components rr’ and tt. Each lies in a focal plane. The converging telescope beam is reimaged to the second focus by intermediate optics that may schematically consist of two ellipsoidal mirrors (matched to apply equal phase shifts to the two beam paths) or similar optics, e.g. pairs of off-axis paraboloids. These auxiliary optics must preserve the high wavefront accuracy needed by FQPM coronagraphy: rms wavefront quality δ, in nm, gives  a starlight rejection factor Rs(δ)=[λ/(πδ)]2. The second beam splitter is mounted upside-down with respect to the first, to impart r’ rather than r to the upper beam path. Two beams emerge from the second beam splitter, but only the upper output port combines the components necessary for on-axis nulling, so half of the incident signal power is lost. (But contrast rather than raw sensitivity is often paramount in coronagraphic observations: even rather bright companions may be lost in the glare of much brighter stars.)
To deliver only correctly-phased beams to the appropriate focal-plane quadrants, intensity masks (blocks) are inserted as shown in Fig. 2. Note that these are not the sophisticated phase-shifting masks of thin-film FQPM devices considered earlier: instead, they are simple amplitude stops, opaque and clear in alternating quadrants. The two masks are clocked with respect to each other by 90°, as indicated in Fig. 2, and this directs rr’/tt/rr’/tt to the four quadrants, giving phases alternating by π as needed for FQPM coronagraphy.
4. Matching intensity between quadrants; fabrication and alignment issues
To realize FQPM coronagraphic operation, the focal-plane mask should ideally give uniform intensity over the four quadrants at the output, where the rr’ and tt beams reunite. In the approach presented here, that requires careful beam splitter design and fabrication to ensure matching of R=T as closely as possible over a broad spectral range. However, although the sensitivity of FQPM operation to intensity imbalances due to tip/tilt residuals is great , it appears that sensitivity to imbalances of the kind incurred by the proposed beam splitter phase shifting scheme is much less. A beam splitter with imperfectly matched R=T will still give uniform intensity over each quadrant, and will match intensities in paired (opposite) quadrants. Quadrants I and III will not match quadrants II and IV, however. This still gives a moderate nulling of the excess light, analogous perhaps to the partial nulling of a stellar image placed on a quadrant boundary but not at the center of an FQPM focal-plane mask. This effect substantially relaxes the R=T requirement; performance is much better than would be indicated by assuming that all unmatched light detracts from rejection. As a quantitative example of this, numerical simulations were carried out for beam splitter parameters well within the current state of the fabrication art. Matching R and T has been demonstrated to better than a percentage point over substantially larger bandwidths than, for example, the astronomical K-band (Δλ/λ~0.15) . (The particular example cited was for one linear polarization at a 45° angle of incidence.) Taking 51:49 intensity ratios in an FQPM coronagraph gives a rejection exceeding 1000, depending on the radius of Lyot stop chosen. (Other errors are assumed negligible.) So though made for another purpose, these beam splitters could give competitive FQPM performance if used in the configuration of Fig. 2. Better beam splitter coatings still appear achievable if specifically designed for this application. Phases of transmission and reflection coefficients may be left unconstrained in the design, their coordination automatically assured by Eq. (3), and care may be concentrated on equalizing magnitudes of r and t. One could optimize coatings for just one polarization, and rely on polarizing filters to select only that component through the beam-splitter FQPM.
For high-performance FQPM operation, the intensity masks in Fig. 2 should have edge precision of ~(λ/D)/30 in angular units on the sky, or ~1 um in distance units in the focal plane at K-band (λ=2.2 μm) for the f/16 beam in the Palomar FQPM system . This precision is readily achieved with optical contact lithography. One mask must be optically superposed on the image of the other in the first beam splitter to a comparable precision and stability. This is a fine mechanical adjustment, but feasible. (This tolerance could be eased, of course, by operating the broadband FQPM mask proposed here in a slower beam, of higher f/#.) The difference of optical pathlengths along the two arms between the two focal planes must be stable; for highest performance, active stabilization might be required.
The quadrant-mask mounting scheme shown in Fig. 2 is intended to be somewhat schematic and conceptual, illustrating the principle of beam splitter phase shifts. Detailed design work will be required to find the best layout, and to determine if this approach is fundamentally competitive with the alternatives. Beam angles of incidence are drawn rather arbitrarily at 45°, but may well be optimized at some other angle. Polarization effects at M1, if literally an ellipsoidal mirror, could be minimized by steering beams to near-normal incidence with folding flats. More convenient mask arrangements may be found. Masks must be within roughly a depth of focus, ~±2(f/#)2λ ~±2(f/#)[(f/#)λ], of the beam splitter, measured along the ray path, and they must ideally be at least ~32λ/D in diameter for optimum FQPM nulling . Comparison of these two expressions shows that at f/16 the masks will not interfere with each other or with the incident optical beam if placed as in Fig. 2. (For f/16 at K-band, the mask should ideally be 1 mm long, and placed within ±1 mm of the focal point on the beam splitter; it could be placed as close as 0.5 mm before contact, though these constraints might dictate using a moderately thin beam splitter substrate.) These tolerances also relax if a higher f/# is chosen. Ghost reflections in the beam splitters, which might limit peak PSF attenuation, may be eliminated by using substrates wedged at a few degrees.
FQPM coronagraphy is a promising technique for achieving deep suppression of light from a bright central star, and searching to a very small inner working angle for faint companions. Achromatic phase masks would be most useful for astronomy. This paper has presented a novel scheme exploiting the natural phase relationships that exist in dielectric beam splitters. The design requires matching the R, T coefficients of each beam splitter to moderately high precision over a broad spectral band, but the π phase shifts needed for FQPM operation are achieved automatically for all wavelengths and both polarizations.
The research described in this publication was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
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