Monochromatic verification of high-contrast imaging with an occulter

Dan Sirbu; N. Jeremy Kasdin; Robert J. Vanderbei

doi:10.1364/OE.21.032234

1. Introduction

The problem of direct imaging of an Earth-like exoplanet is challenging because its reflected light is ten orders of magnitude fainter than the light from the parent star, and its angular separation is very small such that it is easily obscured by the wings of the star’s point spread function (PSF) [1]. Developing the capability to image directly a terrestrial planet is a major goal of space astronomy over the coming decades and would be scientifically invaluable as it would allow extracting spectral information from reflected light and detecting the presence of biomarkers such as the vegetation red edge [2].

The earliest proposal for a starlight suppression system was made by Lyman Spitzer [3]; in this landmark paper, he first proposed the broad concept of a space telescope and then went further by discussing the idea that an external occulting disc could be used to block most of the starlight prior to reaching the telescope pupil thus enabling the direct imaging of planets around nearby stars. He realized that diffraction from a circular disc would be problematic for imaging an Earth-like planet due to an insufficient level of light suppression across the telescope’s pupil, which, in addition, features a bright Poisson spot [4]. He posited that a different edge shape could be used instead. A large-surface partially transmissive screen acting as an apodization function can minimize constructive interference across the shadow [5], but cannot be manufactured to precise enough surface quality to ensure an adequate wavefront. Instead, we propose using an occulter with a binary shape, one that either allows or blocks starlight passage. In Vanderbei et al [6], it was shown that N-fold circular symmetry for a binary mask can be used as a very good approximation of the apodization function and forms the basis for petalized hypergaussian [7] and optimized occulter designs [8]. Figure 1(a) illustrates the external occulter concept. Proposed occulter reference designs range from flagship-class to probe-class and feature occulters tens of meters in diameter separated from the telescope by tens of thousands of kilometers. The performance of such a mission [9] is illustrated in Fig. 1(b) and 1(c).

Fig. 1 (a) Schematic of occulter mission concept—the occulter blocks starlight from reaching the telescope pupil, but allows light from the exoplanet. Using the parameters for a proposed, realistic space mission design [9], a numerical simulation shows the suppression performance across the pupil plane (b) and the contrast for the starlight residual point spread function (c). The white circle in (b) shows the location of the 4-m telescope centered in the shadow. The red circle in (c) shows the 75 mas inner working angle of the occulter—planets can be directly imaged outside this angle.

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Occulters are designed using numerical algorithms based upon scalar diffraction theory. A full-scale ground-based verification that these designs meet their design criteria is impossible given the long separation distances required. Direct scaling of such an occulter and telescope to laboratory size is also impossible because it requires a corresponding scaling of wavelength, which would move to the X-ray band. Therefore scaling to laboratory size is done by maintaining equivalent Fresnel numbers, which produces a functionally identical diffraction integral. Previous laboratory tests [10–12] have been primarily limited by three factors: phase aberrations induced by surface errors from optics used to collimate the beam, diffraction effects due to the finite extent of the optical enclosure, and diffraction off the struts used to support the occulter mask. Our experiment addresses these limitations by using an expanding beam to minimize phase aberrations from optical elements, and by using an outer ring designed to minimize ringing effects from sharp edges while providing a diffraction-friendly mounting method using a series of struts.

The paper is organized as follows. In Section 2, we show how scaling by maintaining constant Fresnel numbers allows us to test, in the lab, a diffraction integral which is identical to the space case. In Section 3, we describe the design of the apodized inner occulter, the outer ring, and the annular binary occulter mask used in the laboratory. We also present a circular occulter mask that acts as a baseline measurement for the performance of the optimized occulter. In Section 4, we describe the laboratory setup used to verify occulter performance. In Section 5, we present the optical propagation equations for the binary mask, we discuss their numerical implementation, and present the theoretical results from the optical equations. In Section 6, we provide experimental measurements of contrast for both the optimized occulter mask and for the circular occulter. In Section 7, we compare the theoretical and experimental results and discuss the limitations of our experimental setup. We conclude with Section 8 in which we discuss the importance of laboratory validations of occulters and how these can be used to inform performance verification of full-scale prototypes.

2. Laboratory scaling

Occulters are opaque screens designed to minimize the effects of diffraction so that the shadow cast on the pupil plane of a downstream telescope is as dark as possible. In space, they operate in the near-field, or Fresnel, regime of the Rayleigh-Sommerfeld diffraction integral. For an experimental validation, we must scale from space dimension to laboratory size in such a manner that we maintain the validity of the optical model in which the occulter operates. We must also demonstrate that the occulter shape has a measurable impact on controlling light diffraction.

2.1. From space to laboratory size

In space, a plane wave passes an occulter and propagates downstream with a shadow. Because of diffraction, some light gets into the shadow area. We can design an apodized (or specially shaped) mask to limit the amount of light incursion. In a laboratory, it is impossible to form an incoming beam of infinite extent. The walls, ceiling, floor, table, etc. all obstruct the beam and complicate the downstream propagation. To address this problem, we design an apodized-edged large hole to shine a beam through. The apodization is chosen to mitigate the downstream diffractive effects. An occulter can then be placed at the center of this hole (how such an occulter is supported will be addressed later). By superposition, the diffraction effects of the hole and the occulter are additive.

Therefore, we consider an apodized hole having a transmittance profile A(r) with range [0, 1] with 0 being fully opaque and 1 fully transmissive. The electric field E_ap(ρ) calculated at a distance z downstream from the occulter illuminated by a collimated beam is given by (see [13]):

E_{ap} (ρ) = \frac{2 π}{i λ z} e^{\frac{π i}{λ z} ρ^{2}} \int_{0}^{R} e^{\frac{π i}{λ z} r^{2}} J_{0} (\frac{2 π r ρ}{λ z}) A (r) r d r

Where ρ is the distance off-axis, λ the wavelength of light, and R the maximum radial extent of the partially-transmissive hole, and J₀ is the zeroth order Bessel function of the first kind arising due to the circular geometry.

To scale the occulter from space size to lab size we introduce a scaling factor q such that the Fresnel numbers $\frac{r^{2}}{λ z}$ and $\frac{ρ^{2}}{λ z}$ remain constant by setting r′ = r/q, ρ′ = ρ/q, and z′ = z/q². With this change of variables, the downstream electric field remains unchanged. Letting A′(r′) = A(qr′), it is easy to check that the downstream electric field is the same (after rescaling): E′_ap(ρ′) = E_ap(qρ′). One important limitation of this approach should be noted, namely that while this scaling maintains constant Fresnel numbers and an identical diffraction integral it does, however, mean that the geometric ratios change. Therefore, the inner working angle defined by the occulter edge is increased in the lab compared to what we would expect in space.

2.2. Diverging beam scaling

Previous laboratory testing of occulters has been limited by wavefront aberrations induced by the collimating optics [10–12]. The reason for collimating the input beam is that observation of a distant star is equivalent to a planar wavefront with insignificant phase variations. To address this issue, we introduce a scaling that maintains the diffraction integral in the presence of an expanding beam. An added benefit of this approach is that it expands the size of the dark hole allowing us to further reduce the separation distances to a reasonable laboratory scale.

First, we introduce a scaling factor γ > 1 applied to the two transverse planes in the following manner. The radial distance across the occulter plane is scaled as r″ = r′/γ, whereas the radial distance across the telescope pupil plane is scaled as ρ″ = γρ′. The separation distance remains unchanged (z″ = z′). This change of variables is illustrated in Fig. 2. We will show that when we perform this change of variables, the diffraction integral can be physically understood as being equivalent with a diverging beam.

Fig. 2 Change of variables resulting in diverging beam scaling. The extremity of the mask under a collimated beam is shrunk by a factor γ, whereas the extremity of the shadow is increased by a factor γ. The divergence angle is given by the proximity h of the source and through similar triangles an expression for γ can be found.

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We let E″_ap(ρ″) = E′_ap(ρ″/γ), A″(r″) = A′(γr″), and R″ = R′/γ and obtain the following propagation equation:

E_{ap}^{″} (ρ^{″}) = \frac{2 π γ^{2}}{i λ z^{'}} e^{\frac{π i}{λ z^{'}} \frac{{ρ^{″}}^{2}}{γ^{2}}} \int_{0}^{R^{″}} e^{\frac{π i}{λ z^{'}} γ^{2} {r^{″}}^{2}} A^{″} (r^{″}) J_{0} (\frac{2 π r^{″} ρ^{″}}{λ z^{'}}) r^{″} d r^{″}

Since γ > 1, we can write γ² = 1 + z′/h, where h is a positive constant (having units of length). Substituting, this expression for the constant γ where it appears in the integral in Eq. (2), we get:

E_{ap}^{″} (ρ^{″}) = \frac{2 π γ^{2}}{i λ z^{'}} e^{\frac{π i}{λ z^{'}} \frac{{ρ^{″}}^{2}}{γ^{2}}} \int_{0}^{R^{″}} e^{\frac{π i}{λ} \frac{{r^{″}}^{2}}{z^{'}}} e^{\frac{π i}{λ} \frac{{r^{″}}^{2}}{h}} A^{″} (r^{″}) J_{0} (\frac{2 π r^{″} ρ^{″}}{λ z^{'}}) r^{″} d r^{″}

Next consider a spherical input beam:

E_{in}^{″} (r^{″}) = \frac{1}{\sqrt{h^{2} + {r^{″}}^{2}}} e^{\frac{2 π i}{λ} \sqrt{h^{2} + {r^{″}}^{2}}}

\approx \frac{1}{h} e^{\frac{2 π i}{λ} h} e^{\frac{π i}{λ} \frac{{r^{″}}^{2}}{h}}

Equation (5) is the second-order Taylor series approximation of the precise form of the spherical input beam in Eq. (4). For typical laboratory sizes we have h ≫ r″, which means the amplitude is almost constant across the occulter mask.

Thus, we can interpret Eq. (3) as the Fresnel propagation of a rescaled occulter mask with a diverging input beam. We also observe that the equivalent Fresnel number F_eq across the occulter is now the summation of the rescaled occulter radius and the finite distance of the diverging beam. The equivalent Fresnel number is equal to the Fresnel number of the collimated occulter:

\begin{array}{l} F_{eq} & = F_{oc} + F_{div} \\ = \frac{{R^{″}}^{2}}{λ} (\frac{1}{z^{'}} + \frac{1}{h}) \\ = \frac{{R^{'}}^{2}}{λ} \frac{1}{z^{'}} \\ = F_{orig} \end{array}

To summarize, the scaling approach we used is to first scale the occulter according to the laboratory separation distance as shown in the previous section, and to then re-scale the occulter mask to account for the expanding beam.

3. Mask design

The experiment uses two different masks, an optimized occulter mask and a circular control mask. We describe the design and manufacturing of these masks.

3.1. Occulter mask

To create an occulter mask in the lab associated with the inner apodization profile, we first design an occulter for space dimensions which we shrink to laboratory size as discussed in the previous sections. Thus, the inner apodization profile, which corresponds to the occulter, is identical in space and in the laboratory, except for a rescaling of the radial variable. We separately design an outer ring that minimizes the effect of diffraction in the central part of the shadow. When the inner occulter is combined with the outer ring, this creates a mask with an annular transmissive region.

3.1.1. Inner occulter

To design the inner occulter screen, we formulate, as first proposed in [8, 14], an optimization problem that minimizes an objective function defined as the suppression. We denote the electric field due to the apodized occulter screen as E_occ, and use Babinet’s Principle [15] to facilitate computation of the diffraction integral to infinity by writing this expression in terms of a complementary apodized hole with apodization Ā_inn(r):

E_{occ} (ρ) = 1 - \frac{2 π}{i λ z} e^{\frac{π i}{λ z} ρ^{2}} \int_{0}^{R} e^{\frac{π i}{λ z} r^{2}} {\bar{A}}_{inn} (r) J_{0} (\frac{2 π r ρ}{λ z}) r d r

We pose constraints independently on the real and imaginary parts of the electric field at the pupil plane of the shadow. This is a conservative approximation of the full, quadratic problem. The radial apodization profile of the inner occulter is the complement of the apodized hole Ā_inn(r), which is discretized appropriately, and represents the set of decision variables which the optimization outputs. The optimization problem can therefore be written as follows:

\begin{array}{l} min : & c \\ subj . to : & - \frac{c}{\sqrt{2}} \leq Re (E_{occ} (ρ; λ)) \leq \frac{c}{\sqrt{2}} \\ - \frac{c}{\sqrt{2}} \leq Im (E_{occ} (ρ; λ)) \leq \frac{c}{\sqrt{2}} \\ s \geq 0 \\ {\bar{A}}_{inn} (r) = 1, 0 \leq r \leq a_{inn} \\ \frac{d {\bar{A}}_{inn} (r)}{d r} \leq 0, | \frac{d^{2} {\bar{A}}_{inn} (r)}{d r^{2}} | \leq σ, \forall 0 \leq r \leq R_{inn} \end{array}

In the problem statement in Eq. (8), we have imposed constraints in the form of smoothness and monotonicity conditions on the apodization profile, and a central opaque disk; σ represents the smoothness condition threshold, a_inn the extent of the opaque central disk, R_inn the extent of the inner occulter, c is the suppression, and [λ_min, λ_max] defines the shadow suppression band. Additionally, a minimum feature size is imposed for manufacturing reasons. The inner occulter apodization profile output (A_inn = 1 − Ā_inn) from the optimization is shown in Fig. 3(a), where the constrained opaque circular disk a_inn and the maximal extent of the occulter R_inn are labelled.

Fig. 3 Resulting optimized apodization profile ouput for the (a) inner occulter (b) outer ring.

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3.1.2. Outer ring

There are a number of challenges associated with testing an occulter in the lab. One concern is mounting the mask in a way such that the supports used do not diffract light into the shadow, an effect which was observed when even very thin fishing lines were used in previous occulter experiments [12]. A second concern is the finite extent of the occulter enclosure, whereas in space the occulter is surrounded by the infinite extent of open space. Our approach for mitigating both of these concerns is to design an outer ring that limits such diffraction effects.

Mathematically, we seek to minimize the difference between the electric field due to the occulter only and the complete annular apodization profile. Letting E_Δ = E_occ − E_ap, where E_ap is the Fresnel integral due to the entire apodized annular mask expressed in Eq. (1). We can now write an outer ring optimization problem similar to that in Eq. (8) by placing constraints on the real and imaginary parts of E_Δ. We can also specify a constraint that forces a fully transmissive hole over the radius a_out and also specify the radial extent of the ring R_out. The output apodization profile A_out minimizes the difference between the desired occulter shadow and the actual shadow in the presence of the outer ring; this is shown in Fig. 3(b) with the fully transmissive region a_out and the outer radial extent R_out labelled.

3.1.3. Combined mask

We combine the apodization of the inner occulter A_inn with that of the outer ring A_out to create the final annular apodized profile A for the occulter mask. This is shown in Fig. 4(a) rescaled to laboratory size by a change of radial variables while maintaining the same apodization profile to that in space. The entire apodization profile is multiplied by a factor b = 0.9 that only affects overall throughput for the apodized case. When turned into a binary shape as shown in Fig. 4(b), this results in a set of struts that cover 10% of the annular area that can support the inner occulter mask and which, with a sufficiently large number of petals, do not theoretically affect the electric field in the dark hole.

Fig. 4 (a) A comparison of the laboratory apodization profiles for the optimized occulter mask and for the control mask. (b) Sixteen petal binary realization of occulter mask (c) Picture of the silicon-etched occulter mask used in the laboratory. (d) Circular occulter baseline mask with sixteen support struts.

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The binary mask is created from N = 16 petals that are defined by turning the apodization profile into the edge of the binary mask [6], with opaque points on the mask belonging to the set S:

S = {(r, θ), 0 \leq r \leq R, θ \in Θ (r)}

Θ (r) = \cup_{n = 0}^{N - 1} [\frac{2 π n}{N} - \frac{π}{N} A (r), \frac{2 π n}{N} + \frac{π}{N} A (r)]

The occulter mask we use is shown in Fig. 4(c). It was manufactured at the Microdevices Lab at JPL using Deep Reactive Ion Etching (DRIE) to etch the open areas of the occulter mask [16,17]. The silicon wafer is 101.6 mm in diameter and 400 μm in thickness. The DRIE process results in vertical etches of the sidewall and thus two steps were used to thin the sidewall to 50 μm. A 100 nm titanium coating was applied to increase the optical absorption to several orders of magnitude beyond the expected level of light in the shadow of the mask. Spot microscope imaging of the resulting mask has shown differences from the theoretical shape at the petals, which are the most challenging manufacturing features, to have a standard deviation of 2.5 μm.

3.2. Control mask

To establish a baseline level to verify the performance of the occulter mask, a simple control mask was manufactured that consists of a circular occulter with supporting struts to a circular outer edge. The transmission profile corresponding to the control mask consists of an annular region starting at a radius that corresponds to the outer tip of the inner petals of the high-contrast mask and extending to a radius that corresponds to the outermost open part of the high-contrast mask. This transmission profile is then multiplied by the same scaling factor b = 0.9 as for the occulter mask to create support struts. The resulting apodization profile is shown in Fig. 4(a). The same petalization process described by Eq. (9) and (10) for the occulter mask is applied to obtain a binary realization of the control mask, and although the inner and outer edges are simply circular this apodization produces sixteen support struts as shown in Fig. 4(c) by comparison with the optimized profile. The region between the inner and outer circles defines the annular observation region for the control mask. The control mask was manufactured by etching the open areas from a rectangular piece of copper. Copper etching has much less accurate edge features than the DRIE method used for the optimized occulter mask; however, the expected level of light in the shadow is greater hence significantly relaxing the tolerance on the edge features.

4. Experimental setup

The occulter testbed, located at Princeton University, consists of a large enclosure 40′ × 4′ × 8′ that blocks ambient light. There are two passively isolated Newport optical tables at each end of the enclosure. The larger optical table contains the optics that create an artificial star, which, as explained in section 2.2 consists of a diverging beam. The smaller optical table at the opposite end of the enclosure contains the telescope optics for observarvation of the artificial star and consists of a camera mounted on two long-travel stages that provide two degrees of freedom of movement in the lateral and vertical directions. The layout is shown schematically in Fig. 5.

Fig. 5 Layout of the Princeton Occulter Testbed. On the left-end of the enclosure is the optical table containing the artificial star and on the right-end is the moveable telescope.

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The light source is a single-mode 2mW HeNe laser operating monochromatically at 632 nm. To create the diverging beam, the beam is passed through two lenses acting as a beam expander, and focused onto a 15 μm pinhole through an off-axis parabolic mirror. We use a diverging beam through a pinhole so that the pinhole acts as a spatial filter that removes high-frequency aberrations due to surface error on the optics. The choice of lenses and pinhole affects the divergence angle of the beam. The diverging beam propagates 1.5 m before encountering the occulter mask which it overfills at the end of the optical table. Baffles are placed around the occulter mask extending to the walls of the enclosure and a further two sets of full wall baffles are located downstream to ensure no light propagates to the camera around the annular mask and to minimize any scattered light. The mask is tilted 5° to eliminate a ghost reflection by directing the back-reflection into a black foil beamdump.

Beyond the occulter mask, the beam propagates 9.1 m to the second optical table on which a camera is placed on two 300 mm long-travel stages; both move perpendicular to the propagation direction, one horizontally and one vertically. Together these stages allow the camera to be precisely aligned in the shadow cast by the occulter. The camera is an astronomical-grade thermoelectrically cooled Starlight Xpress SXV-H9 CCD. A telephoto lens set at f = 300 mm is outfitted via the M42 mount, and the smallest available aperture setting at f/22 is used to form a six-bladed iris—at a diameter of 14 mm this is smaller than the designed dark hole (see Table 1). The CCD has pixel pitch of 6.45 μm × 6.45 μm, which, due to the scaling to lab size, results in a spatial resolution more than an order of magnitude better than expected in space for 18 mas pixels [9]. The increased spatial resolution in the laboratory allows us to better resolve the point spread function and identify any unexpected sources of light.

Table 1. Summary of experimental parameters and their equivalent space scaling.

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Two key metrics are used for measuring the performance of an occulter. Suppression is measured in the pupil plane and represents the ratio of the amount of light in the mask’s shadow to the amount of light without the mask. This measurement is taken by first centering the camera in the shadow of the occulter mask without the lens attached and taking long exposures; then the mask is removed and short exposures are retaken. Contrast is measured by reattaching the camera lens and focusing on the pinhole source. The contrast is the ratio between the intensity of each pixel in the image formed when the mask is in place and the intensity of the peak pixel of the point spread function without a mask, calibrating for different exposures. The occulter is designed to obtain maximal suppression, however the measurements described in this experiment are taken in the image plane and measure contrast directly. When performing a diffractive simulation with an increased spatial resolution in our laboratory comparable to that expected for a realistic space mission, the contrast performance is theoretically similar because the central pixel encloses a similar amount of the total energy.

5. Theoretical analysis

To determine the theoretical performance of both the control and the occulter masks we use a numerical simulation that measures the contrast similarly to the experimental method.

5.1. Pupil plane propagation

We first need to propagate the input electric field past the occulter mask to the plane of the camera. The occulter mask is designed by using the Fresnel propagation in Eq. (1) which corresponds to a partially transmissive mask A(r) exhibiting full circular symmetry. However, the occulter mask is petalized with points on the occulter belonging to set S described by Eq. (9) and (10). The resulting Fresnel propagation for an input beam E_in(r) across the occulter plane is:

E_{bin} (ρ, ϕ) = \frac{1}{i λ z} \int \int_{S} E_{in} (r) e^{\frac{π i}{λ z} (r^{2} + ρ^{2})} e^{\frac{- 2 π i}{λ z} r ρ cos (θ - ϕ)} r d r d θ

To numerically propagate the expanding beam past a petalized mask, we take advantage of the N-fold circular symmetry of S. We apply the Jacobi-Anger expansion given by the expression:

e^{- i \frac{2 π r ρ}{λ z} cos (θ - ϕ)} = \sum_{m = - \infty}^{\infty} i^{m} J_{m} (\frac{- 2 π r ρ}{λ z}) e^{i m (θ - ϕ)}

This allows us to re-write Eq. (11) by separating the angular and radial integrals. Performing the angular integral results in all terms disappearing except those for which m = ±kN. Taking advantage of the evenness of Bessel-functions allows us to write the propagation of the electric-field past the petalized mask as:

\begin{array}{l} E_{bin} (ρ, ϕ) & = \frac{2 π}{i λ z} \int_{0}^{R} E_{in} (r) e^{\frac{π i}{λ z} (r^{2} + ρ^{2})} J_{0} (\frac{2 π r ρ}{λ z}) A (r) r d r + \\ \dots \sum_{k = 1}^{\infty} \frac{4 π cos (k N ϕ)}{i λ z} \int_{0}^{R} E_{in} (r) e^{\frac{π i}{λ z} (r^{2} + ρ^{2})} J_{k N} (\frac{2 π r ρ}{λ z}) \frac{sin (k π A (r))}{k π} r d r \end{array}

This expression is essentially the original circularly-symmetric Fresnel propagation in Eq. (1) with the angular dependence at the pupil plane provided by an expansion of higher-order Bessel terms. This expansion converges quickly as more terms are added. We use K = 20 terms. We use the apodization profiles for the control and occulter masks as shown in Fig. 4(a). The input beam is normalized to unity amplitude. The apodization profile used in the simulation is the identical theoretical apodization profile output from the optimization with 8000 points across the occulter plane. A sensitivity analysis was performed using a spline interpolation to increase the sampling by two times across the occulter plane, and results were consistent with the original sampling. The electric field at the pupil plane is sampled at 2000 by 2000 points across the pupil function as described in the next section, and is obtained by nearest-neighbour interpolation from the Bessel terms.

5.2. Image plane propagation

We let (x, y) be the Cartesian coordinates across the pupil plane. E_pup(x, y) describes the electric field at the pupil plane. We define a circular pupil function p(x, y):

p (x, y) = {\begin{array}{l} 1, & \sqrt{x^{2} + y^{2}} \leq r_{a} \\ 0, & else . \end{array}

where r_a is the radial extent of the imaging lens. Thus, for a lens of f = 300mm operating at f-stop f/22, r_a = 6.8mm. Assuming the imaging optic is a thin lens, to focus on a distance d in front of the lens we Fresnel propagate a distance s past the lens which introduces a quadratic phase function. Then the expression for the electric field at the image plane defined by Cartesian coordinates (u, v), ignoring leading phase terms, becomes:

E_{im} (u, v) = \frac{1}{i λ s} \iint_{p (x, y) = 1} E_{pup} (x, y) e^{\frac{- π i}{λ d} (x^{2} + y^{2})} e^{\frac{- 2 π i}{λ s} (x u + y v)} d x d y

This expression can be evaluated as a matrix Fourier transform [18]. We match the experimental setup by selecting the same pixel pitch of the science camera, a radial extent that matches the f-stop, and by choice of the propagation distance s and focal distance d. The simulation parameters are summarized in Table 2.

Table 2. Summary of simulation parameters.

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The image plane simulations for the control and occulter mask are shown in Fig. 6(a) and (b) respectively. To obtain the contrast calibration, the propagation to the image plane is carried out for an apodization profile consisting of a mask with an apodization profile that is fully transmissive up to the radial extent of the mask holder. The contrast is obtained from the ratio with the highest intensity pixel in the image plane for this point spread function. For the control mask in Fig. 6(a), a 2 mm horizontal displacement is introduced in the camera’s location. This displacement in simulation eliminated the symmetry of diffraction features due to which the model predicted a bright spot at the centre of the image plane, but this was never seen in measurements.

Fig. 6 Simulation of contrast in the image plane (a) using the control mask (b) using the occulter mask. The inner and outer red circles denote the annular dark region for both masks.

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6. Experimental results

Experimental data is collected by stacking a series of images with identical exposures taken when the camera is aligned with the centre of the shadow cast by the optical mask. The exposure time is chosen to maintain the camera’s pixels in their linear regime. To reduce dark noise and camera bias from the stacked frame, for every set of frames collected a set of darks is taken with the same amount of camera exposure but with the artificial star turned off. These dark sets are median-combined and subtracted from each measurement frame to eliminate signals from unwanted sources, such as dark current.

Actual combined image-plane results are shown in Fig. 7. The occulter mask dataset consists of 10,000 frames collected at 1 sec exposure. To calibrate for contrast, the flux in each pixel of the stacked frame is divided by the peak pixel flux when the mask is removed. To prevent over-exposure in the calibration frames, two OD2.0 filters were placed in series before the pinhole. The two OD2.0 filters have measured transmissions of 1.00 ± 0.01% and 1.05 ± 0.02%. The stacked calibration dataset consists of 100 frames each taken at 0.01 sec to minimize the effect of power fluctuations in the laser source.

Fig. 7 Contrast measurements at the image plane (a) using the control mask and (b) using the occulter mask. The inner and outer red circles denote the annular dark region for both masks.

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In Figure 7(a) the contrast calibrated image plane measurement of the control mask is shown. The control mask dataset consists of 10,000 frames collected at 0.05 sec exposure. Short exposures were necessary to prevent saturation of the bright features of the image while a large number of frames was used to achieve a longer overall integration time allowing features to be seen in the darkest part of the annular region. In Figure 7(b) the contrast calibrated image plane measurement of the optimized occulter mask is shown.

7. Discussion

The main purpose of the control mask is to demonstrate the accuracy of the calibration method for computing contrast—that is, that the assumptions behind the propagations to the image plane such as focal distance match the experimental results. On a qualitative level, we compare the contrast-calibrated image-plane the theoretical in Fig. 6(a) with the measurement in Fig. 7(a). The scalar diffraction simulation captures all the main features of the experimental image: two bright inner and outer rings with sixteen divisions at the point of attachment of the support struts, and fine diffraction ringing around the main bright rings. For a quantitative analysis, we provide a cross-section through the image plane and plot the azimuthal median as shown in Fig. 8(a). We also plot a 95% confidence interval for the median based on percentile populations [19]. The magnitude of the inner peaks are within a factor of 1.06 of each other and for the outer peaks within a factor of 1.56. This relatively close agreement demonstrates the validity of the calibration method and scalar diffraction simulation as we are able to predict contrast of the bright rings.

Fig. 8 Azimuthal cross-sections through the image plane of (a) the control mask (b) occulter mask comparing the theoretical median with the measured median and its 95 % confidence interval. The dotted vertical lines represent the inner and outer working angles of the annular openings and the 50 % throughput point for the occulter mask (c) Location of the wedge areas used to create the azimuthal cross-section for the occulter mask shown with lighter hue.

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A qualitative analysis for the optimized occulter mask between the theoretical in Fig. 6(b) and the measurement in Fig. 7(b) shows some significant differences. In the diffraction simulation the struts are not expected to be bright, but in the experimental image the struts are glowing. Some tips of the outer petal are brighter than others, which can be attributed to the tilt of the mask to remove the ghost reflection (changing the tilt of the mask predictably changes the bright tips). Lastly there is a glow around the inner mask which is both radially larger and two orders of magnitude brighter than the diffraction analysis predicts. To obtain a quantitative measure of the performance of the optimized occulter mask, we plot a similar cross-section plot using the azimuthal median in Fig. 8(b). To avoid biases due to the bright struts, we choose a set of 16 wedges whose location and size are chosen to avoid the struts as shown in Fig. 8(c). Thus, the azimuthal median in Fig. 8(b) is computed using the signal contained within the wedges only and shows an improvement of contrast performance over the control mask of about two orders of magnitude. The median contrast across the wedges at the inner working angle of 400 mas space-equivalent is 1.05 × 10⁻¹⁰ and this improves towards the outer working angle at 638 mas space-equivalent to 2.51 × 10⁻¹¹. Nonetheless, the optimized occulter mask performs worse than the theoretical diffraction analysis by about two to three orders of magnitude. The bright inner and outer features, which were in close agreement for the control mask are also brighter than expected.

Figure 9 has three panels which allow for easier comparison of the control mask with the occulter mask. In Figure 9(a) the experimental control mask result is shown again, while in Fig. 9(b) this same plot has the log-scale restretched to match the experimental occulter mask result in Fig. 9(c). The mean contrast is calculated in four regions of interest, and it is shown that the struts of the control mask are at a similar level to those of the occulter mask but are less visible than those of the occulter mask. The reason for the reduced visibility of the struts is that diffraction limits the contrast in the annular regions of the control to a level similar to the struts themselves, whereas for the occulter mask, the annular regions are about two orders of magnitude below the level of the struts.

Fig. 9 Experimental contrast results for the (a) control mask, (b) control mask using the same log stretch as the (c) occulter mask. Four equal area regions are shown in yellow. The mean contrast across the control mask strut in Region 1 is 6.03 × 10⁻⁹, while in Region 2 which has no strut the mean contrast is 1.02 × 10⁻⁹. The mean contrast across the occulter mask strut in Region 3 is 3.63 × 10⁻⁹ while in Region 4 which has no strut the mean contrast is 3.31 × 10⁻¹¹.

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Figure 10 shows saturated images of the occulter mask taken with two different pupil apertures, with Fig. 10(a) using a machined circular f/22 aperture, and lastly Fig. 10(b) taken with a smaller f/47 circular aperture. Comparing Figure 10(a) with (b) as we decrease the size of the entrance aperture while keeping the same shape, we see that the diffraction rings around the central bright spots increase. Most importantly, these appear to extend into the dark annular regions where they superpose. This indicates that the scattered light from the edges is the limiting factor in verifying contrast to the level predicted by diffraction theory in the dark annular regions.

Fig. 10 Equal log-stretched saturated images with different entrance apertures: (a) f/22 circular, and (b) f/47 circular.

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To predict the effect of the bright edges on the contrast performance of the occulter, especially in the dark annular regions, we introduce approximately 2000 point sources with uniform intensity δ evenly distributed along the geometric edges of the occulter mask. The electric field contributions of N point sources are superimposed mathematically to the electric field previously computed in Eq. (13) at the pupil plane using diffraction analysis:

E_{pup} (x, y) = E_{bin} (x, y) + δ \sum_{j = 1}^{N} e^{\frac{2 π i (\sqrt{{(u_{j} - x)}^{2} + {(v_{j} - y)}^{2} + z^{2}} + \sqrt{u_{j}^{2} + v_{j}^{2} + z_{0}^{2}})}{λ}}

In the above, the phase distance of the point sources is calculated with respect to the pinhole source. Figure 11 shows the glint simulation in (a) and its corresponding cross-section in (b). The value of δ is chosen so that the simulated inner peak matches the measured in the azimuthal cross section. As shown in Fig. 11(b), the simulated median with glint predicts the contrast in the dark annular region for the experimental data. The outer peak from simulation is brighter than the measured peak, which suggests intensity attenuates towards the outer edges of the occulter mask.

Fig. 11 Numerical simulation including glint point sources (a) Contrast shown in image plane limited in the dark annular region by superposition of the PSFs (b) Corresponding azimuthal cross-section using wedges.

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This experiment was performed in monochromatic light using a HeNe laser operating at 632 nm. The occulter mask is optimized to provide broadband suppression over a discrete set of wavelengths spanning the range 400–1100 nm. The specific wavelength of the HeNe laser is between these wavelengths and is thus representative of performance for wavelengths between the optimized wavelengths. We therefore expect that other wavelengths within the optimized suppression band would perform similarly, however this was not tested experimentally.

One concern in this experiment is that the paraxial approximation in Eq. (5) is not accurate to the contrast levels being measured. This is important because the equivalence to the space occulter is based on this approximation. To test this, a simulation of the experiment using a paraxial input beam as shown in Fig. 12(a) is compared to a simulation using a spherical diverging beam. The residual point spread function when subtracting the paraxial point spread function from the spherical is shown in Fig. 12(b). The residual is about an order of magnitude below the original which indicates that for the contrast levels measured in this experiment the paraxial approximation is sufficient.

Fig. 12 (a) Theoretical point spread function using a paraxial input beam (b) Residual point spread function when subtracting from the theoretical spherical beam point spread function.

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8. Conclusions and future directions

In this experiment, we have experimentally verified the contrast performance of an optimized occulter mask to better than ten orders of magnitude in the dark regions of the annular discovery zone in red monochromatic light. We have designed a control experiment which demonstrates the validity of the contrast calibration scheme, and also demonstrates that the optimized occulter does result in improved contrast performance close to that needed to image an exo-earth.

Our experiment has shown some limitations that need to be investigated further in the future. The edges of the occulter mask in image-plane measurements are much brighter than expected from the theoretical simulations based on scalar diffraction. A simulation of glint point sources with uniform intensity chosen so the first peak matched to the experimental data accurately predicts the contrast performance of the occulter mask, demonstrating that the edge brightness is the limiting factor via superposition of the wings of the point spread functions. Microscope spot checks of the edges have shown that the mask is manufactured within micron-level tolerances and simulated edge errors suggest that these are not diffraction effects. The DRIE process results in a final step with a nearly vertical slope. One possible mechanism for this scattered light is that it is due to specular reflection from roughness along the occulter mask’s edge wall due to the diverging beam. Modifications of the mask manufacturing process focusing on sharper, thinner edges could result in a fifty-fold reduction of edge thickness down to 1 μm; furthermore, edges can be coated to reduce their reflectance, but this would have to be achieved while maintaining edge manufacturing precision. A possible mechanism for the measured edges could be surface plasmonic reradiation due to evanescent coupling resonance—however, we are measuring more than two orders of magnitude enhanced transmission which is a significantly higher level than reported elsewhere in literature [20]. Furthermore, placing a narrowband filter with 1 nm transmission at the camera end did not alter the bright features at the mask edges when compared to placing the same filter prior to the occulter mask.

Assuming that the linear density of the observed glints remains constant when scaled up to space size, our simulations indicate that for a typical mission design [9] such a glint level would be below the diffraction point spread function, primarily due to the increased distance between the occulter and the telescope. Nonetheless, it is not clear whether the glint observed in the lab can be scaled up to space size in this manner—for example, in our experiment we use an expanding beam whereas in space the beam is collimated. Furthermore, the glint may be due to environmental factors such as dust particulates increasing the surface roughness of the sidewalls—for a space occulter, zodiacal dust and other particulates could also be a problem. There has been recent work modelling the optical edges of an occulter to predict the amount of scatter from sunlight [21], and this has driven the introduction of requirements for thin, low-radius of curvature edges on the space occulter to reduce scattered light. This work may be used to predict the level of non-diffractive scatter from the target star itself, and the same requirements introduced to minimize sunlight scatter would also reduce starlight specular reflection.

In the future, in addition to verifying contrast for new masks fabricated with sharper edges, our testbed can also be operated at a lower level of contrast where diffraction is the dominating scatter regime. For such a lower-contrast mask, the testbed can be used to demonstrate the alignment of the telescope in the shadow of the occulter using information sampled from the diffractive shadow.

Acknowledgments

We thank Eric Cady and Stuart Shaklan for many helpful discussions, and the reviewers for constructive comments which helped improve the manuscript. This work was partially performed under NASA contract NNX09AB97G and grant 1430187 from the California Institute of Technology’s Jet Propulsion Laboratory. DS acknowledges financial support from the NASA Earth & Space Science Fellowship.

References and links

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2. S. Seager, E. Turner, J. Schafer, and E. Ford, “Vegetation’s red edge: a possible spectroscopic biosignature of extraterrestrial plants,” Astrobiology 5, 372–390 (2005). [CrossRef] [PubMed]

3. L. Spitzer, “The beginnings and future of space astronomy,” Am. Sci. 50, 473–484 (1962).

4. A. Fresnel, Oeuvres completes d’Augustin Fresnel: theorie de la lumiere (Imprimerie imperiale, 1866).

5. C. Copi and G. Starkman, “Big occulting steerable satellite,” Astrophys. J. 532, 581–592 (2000). [CrossRef]

6. R. Vanderbei, D. Spergel, and N.J. Kasdin, “Circularly symmetric apodization via star-shaped masks,” Astrophys. J. 599, 686–694 (2003). [CrossRef]

7. W. Cash, “Detection of earth-like planets around nearby stars using a petal-shaped occulter,” Nature 442, 51–53 (2006) [CrossRef] [PubMed]

8. R. Vanderbei, E. Cady, and N.J. Kasdin, “Optimal occulter design for finding extrasolar planets,” Astrophys. J.665, 794–798 (2007). [CrossRef]

9. N. J. Kasdin, E. Cady, P. Dumont, P. Lisman, S. Shaklan, R. Soummer, D. Spergel, and R. Vanderbei, “Occulter design for THEIA,” Proc. SPIE7440 (2009). [CrossRef]

10. R. Samuele, T. Glassman, A. Johnson, R. Varshneya, and A. Shipley, “Starlight suppression from the starshade testbed at NGAS,” Proc. SPIE7440 (2009). [CrossRef]

11. R. Samuele, R. Varshneya, T. Johnson, A. Johnson, and T. Glassman, “Progress at the starshade testbed at Northrop Grumman Aerospace Systems comparisons with computer simulations,” Proc. SPIE7731 (2010). [CrossRef]

12. E. Schindhelm, A. Shipley, P. Oakley, D. Leviton, W. Cash, and G. Card., “Laboratory studies of petal-shaped occulters, ” Proc. SPIE6693 (2007). [CrossRef]

13. J. Goodman, Introduction to Fourier optics (Roberts & Co., 2005).

14. E. Cady, L. Pueyo, R. Soummer, and N.J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE7010 (2008). [CrossRef]

15. M. Born and E. Wolf, Principles of optics (Cambridge University Press, 1999).

16. E. Cady, K. Balasubramanian, M. Carr, M. Dickie, P. Echternach, T. Groff, N.J. Kasdin, C. Laftchiev, M. McElwain, D. Sirbu, R. Vanderbei, and V. White, “Progress on the occulter experiment at Princeton,” Proc. SPIE7440 (2009). [CrossRef]

17. E. Cady, K. Balasubramanian, M. Carr, M. Dickie, P. Echternach, N.J. Kasdin, S. Shaklan, D. Sirbu, and V. White, “Broadband suppression and occulter position sensing at the Princeton occulter testbed,” Proc. SPIE7731 (2010). [CrossRef]

18. R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15(24), 159345 (2007).

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20. A. Krishnana, T. Thio, T. Kim, H. Lezec, T. Ebbesen, P. Wolff, J. Pendry, L. Martin-Moreno, and F. Garcia-Vidal., “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun.200 (2001).

21. S. R. Martin, S. Shaklan, S. Crawford, S. Lee, B. Khayatian, D. Hoppe, E. Cady, and P. D. Lisman, “Starshade optical edge modeling, requirements and laboratory tests,” Proc. SPIE8864 (2013).

Parameter	Laboratory	Space Equivalent
Pinhole-to-occulter distance	1.5 m	Infinity
Occulter-to-camera distance	9.1 m	97,000 km
Radius of dark zone	13 mm	12 m
Occulter diameter	45 mm	380 m
Outer occulter diameter	89 mm	N/A
Inner working angle	8.4 arcmin	400 mas
Outer working angle	17 arcmin	N/A
Telescope diameter	14 mm	17 m
Wavelength suppression band	400–1100 nm	400–1100 nm

Symbol	Parameter	Value
r_a	entrance pupil radius	6.8 mm
f	focal length of lens	300 mm
s	distance to pinhole	10.7 m
d	distance to image	309 mm
Δu, Δv	pixel pitch	6.45 μm
Δr	occulter midpoint distance	5.45 μm
Δx, Δy	pupil midpoint distance	9.63 μm
K	number of nonzero Bessel terms	20
N	number of petals	16

Parameter	Laboratory	Space Equivalent
Pinhole-to-occulter distance	1.5 m	Infinity
Occulter-to-camera distance	9.1 m	97,000 km
Radius of dark zone	13 mm	12 m
Occulter diameter	45 mm	380 m
Outer occulter diameter	89 mm	N/A
Inner working angle	8.4 arcmin	400 mas
Outer working angle	17 arcmin	N/A
Telescope diameter	14 mm	17 m
Wavelength suppression band	400–1100 nm	400–1100 nm

Symbol	Parameter	Value
r_a	entrance pupil radius	6.8 mm
f	focal length of lens	300 mm
s	distance to pinhole	10.7 m
d	distance to image	309 mm
Δu, Δv	pixel pitch	6.45 μm
Δr	occulter midpoint distance	5.45 μm
Δx, Δy	pupil midpoint distance	9.63 μm
K	number of nonzero Bessel terms	20
N	number of petals	16

Monochromatic verification of high-contrast imaging with an occulter

Abstract

1. Introduction

2. Laboratory scaling

2.1. From space to laboratory size

2.2. Diverging beam scaling

3. Mask design

3.1. Occulter mask

3.1.1. Inner occulter

3.1.2. Outer ring

3.1.3. Combined mask

3.2. Control mask

4. Experimental setup

5. Theoretical analysis

5.1. Pupil plane propagation

5.2. Image plane propagation

6. Experimental results

7. Discussion

8. Conclusions and future directions

Acknowledgments

References and links

Cited By

Figures (12)

Tables (2)

Equations (16)

Optics Express