2.1

Science Operations

Seventy targets have been identified as candidates for the CDEEP survey. Figure 2 shows their L_IR/L⋆ distribution in comparison to all known disks within 300 pc and to those resolved with HST or XAO. The observable CDEEP sample spans the full range of known IR excesses, a wide range of disk sizes, and stars with spectral types evenly divided between early and solar-like stars, at a median age of ~300 Myr (from the literature) and a median distance of ~30 pc. The total number is only limited by target star magnitude and whether the disk size falls within the high-contrast FOV (0.5–6”). The majority of the disks (85%) will be resolved by at least 10 resolution elements (resel, assuming one resel of 0.32”), sufficient for searching for resonant structures perturbed by unseen planets, and bridging the transition region between transport- and collisionally-dominated regimes.¹⁴

The CDEEP sample includes many nearby iconic debris systems such as Vega, Fomalhaut, e Eri and β Pic where complex disk structures have been intensively studied in thermal IR and some in scattered light. They provide a crucial foundation for interpreting debris disk behavior in terms of the underlying planetary system configuration. Excepting β Pic, the inner regions of these benchmark debris systems currently lack scattered-light imaging. Several of the CDEEP targets have NASA/LBTI measurements with exozodi having been detected in the habitable zone for seven of them.¹³ A subset of CDEEP targets are also known to host exoplanets either by RV and/or direct imaging measurements, and this subset is particularly important for studying planet-disk interactions.

2.2

Telescope and Instrument Payload

CDEEP is proposed to be a 34.9 cm off-axis monolithic silicon carbide (SiC) telescope integrated with the coronagraph optical bench, as seen in Figure 3. CDEEP leverages the heritage of the MEMS Deformable Mirror Demonstration Mission (DeMI),^{15, 16} the PICTURE series balloon¹⁷ and sounding rocket missions,^18–20 EXCEDE laboratory testing,^{21, 22} and a wealth of theoretical and laboratory work on VVC technology.^23–27 The unobscured off-axis design eliminates diffraction effects from a conventional secondary obscuration and support structure, significantly easing the challenge of high-contrast imaging.²⁸ The telescope and coronagraph optical layouts are shown in Figure 4. The fast Cassegrain design was chosen to maximize the aperture in the volume, is well matched to the coronagraph size, and can be mounted on a common optical bench, greatly simplifying the metering structuring and minimizing the number of interfaces that must be tracked. At the telescope focus, a reflective pin-hole aperture provides a large FOV acquisition camera imaging for coarse acquisition by passing only the on-axis beam. Fine pointing will be corrected by the Fast Steering Mirror (FSM) with a piezoelectric stage using light rejected by the polarizing beamsplitter between the FSM and OAP1.

Figure 3:

Optomechanical rendering of the unobscured SiC telescope and integrated optical bench (rays from the left).

Figure 4:

Details of the CDEEP coronagraph optical path (from the M2 on the left) using VVC, LP (Linear Polarizer), and QWP (Quarter Wave Plate).

The stabilized on-axis beam will reflect off a deformable mirror (DM) within a relay mirror configuration creating a conjugate plane to the system stop M1. Low-order aberrations are controlled by the DM with feedback from the LLOWFS after the VVC downstream. The “corrected” complex field is focused to a charge-6 Vector Vortex Coronagraph (VVC)²³ using a slow focusing mirror providing a ~f/60 beam at the center of the phase mask. A charge-6 VVC balances relative insensitivity to wavefront errors (Figure 6) with small IWA=2.4A/D. A linear polarizer followed by a quarter wave plate (QWP) circularizes light into the VVC and a downstream (QWP) and linear polarizer reject non-circularized leakage. A single, circular polarization state incident improves broadband VVC contrast²⁹ and allows polarimetric measurements. The phase-modulated beam then propagates to an over-sized OAP mirror, capturing all the diffracted light field from the on-axis star. A reflective Lyot stop (RLS) is placed between the over-sized OAP mirror and the science camera. Off-axis debris disk light passes through the RLS while the on-axis starlight is reflected towards the LLOWFS.

3.1

Optical Design Constraints

Our prototype design was motivated by the goal of demonstrating a coronagraphic testbed that can accurately simulate the behavior of our proposed flight model while minimizing cost. Thus we required:

3.1.1

Talbot Effect Control

Adopting the approach of N’Diaye et al.³⁰ we consider the ramifications of the Talbot effect on the distances between pupil planes. The Talbot effect is a consequence of 3-beam diffraction on sinusoidal gratings³¹ that results in the self-imaging of the grating at particular distances. For a generally rough optical surface, each spatial frequency across the optic acts like a grating with a different associated Talbot distance. In between the grating and its self-image mixing between amplitude and phase is observed, which is a limiter to high contrast imaging instrument performance.³² For small pupil sizes (such as those on MEMS DMs) the Talbot effect occurs at relatively short distances, on the order of the specified testbed footprint. Consequently, the distances between optics must be constrained with consideration for the Talbot effect so that minimal mixing between amplitude and phase will occur. We require our testbed distances to be constrained to less than a quarter of a Talbot length (<< z_t/4) to mitigate the coupling of phase into amplitude. The Talbot distance (z_t) at a particular frequency (f_max) is given by:

Where D is the size of the pupil and λ is wavelength. To meet the required contrast of the testbed it is desireable to control 12 cycles per pupil at a wavelength of 532nm.

3.2

Contrast Requirements

The design of the testbed is also motivated by CDEEP’s contrast requirements, shown in Table 1. CDEEP’s overall raw contrast requirement is 10⁻⁷ per resel, with a post-processing contrast of 10⁻⁸. We have applied substantial conservatism in developing our contrast budget which combines all of the instrument error terms into a raw contrast per resel. The contrast budget is informed by lab tests,^{26, 33} sub-orbital-missions,^{17, 34} and future mission concepts.^{25, 35, 36} Compared to these missions, CDEEP has a smaller aperture and thus less sensitivity to pointing and manufacturing errors. The CDEEP primary mirror is of long focal length and thus is less sensitive to polarization aberrations. In Section 4 we describe and present the results of simulations validating those terms which were not identified from literature for the testbed.

4.1

Sensitivity Analysis

To meet the contrast requirements outlined in Table 1, CDEEP’s wavefront sensing and control (WFSC) system must be able to correct for a variety of dynamic and static aberrations. CDEEP’s overall WFSC layout is shown in Figure 5. The relevant parts of this system for the prototype testbed will be the fine pointing performed by the FSM, the low-order corrections performed by the LLOWFS, and the high-order corrections performed by the DM and the science camera, which we describe below.

Figure 5:

The CDEEP WFSC control system layout includes a high-rate FSM, a high-order MEMS DM fed by a LLOWFS and EFC from the science camera.

First we conducted an analysis of the general sensitivity of the VVC to aberrations in the incoming wavefront, with the aberrations decomposed using Zernike polynomials. An HCIpy³⁸ model of a vector vortex coronagraph was created, and individual Zernike modes were propagated to the image plane for a range of amplitudes. The average contrast in the dark hole region was retrieved, and this contrast as function of aberration amplitude for 23 Zernike polynomials starting from tip can be seen in Figure 6.

Figure 6:

Sensitivity of a charge-6 vector vortex coronagraph to Zernike aberrations.³⁹

Additionally, sensitivity analyses are useful tools in understanding the geometric wavefront error induced by imperfect assembly and alignment of an optical system. Conventional tools such as Zemax OpticStudio allow sensitivity analysis of performance degradation due to perturbations in six spatial degrees of freedom, but only for typical performance metrics like RMS wavefront error, spot size, etc. To inform the mechanical mounting tolerances of the testbed optics for a VVC coronagraph, a numerical model of the coronagraph in HCIpy³⁸ allowed mapping between contrast due to individual Zernike modes induced in the testbed by misalignment errors.

This mapping was used to convert Zernike coefficient values to contrast degradation. To analyze the contrast degradation as a function of tolerance perturbation, a new sensitivity analysis tool was created using OpticStudio’s ZOS-API with Python. The six spatial degrees of freedom (x/y/z decenter, tilts) were perturbed individually, and at each step Zernike coefficients were fit to the resultant wavefront. Multiplying the retrieved Zernike coefficients by the data in Figure 6 yields a map of tolerance perturbation to contrast degradation. Figure 7 shows the sensitivity of M3 with an x-axis chosen for a tip/tilt mount with a resolution of 0.4° per alignment knob rotation. A quarter turn of the knob corresponds to a 0.1°, which is used as the magnitude of displacement in the X/Y tilt plots (lower left, right, respectively). The RSS error is < 10⁻⁸ at the most extreme tilt, indicating this as a suitable mount for tip/tilt control.

Figure 7:

Zernike decomposed sensitivity analysis for the coefficients corresponding to tip, tilt, and the third order monochromatic aberrations. The characteristic feature of this analysis is the gullwing plot shape with a minimum near zero. The range of the × axis of each plot is selected for comparison against the mechanical tolerance of mounting hardware. These plots are used to analyze if a mount will be precise enough to meet the testbed’s static contrast requirements (~ 10⁻⁸). The RSS (Root Sum Squared error, shown in Red) is plotted with the Zernike coefficients to show the total error contribution.

Decomposing the optical element sensitivity into Zernikes also allows consideration for which errors can be corrected using the adaptive optics of the system. For example: static tip (Z2) and tilt (Z3) errors can be compensated for by setting a new baseline tilt on the FSM.

4.2

Thermal Deformation

High-contrast space coronagraphs require sub-nanometer level stability to suppress starlight over extended integrations. Sufficient knowledge of how the optics deform in response to temperature changes is necessary for modeling the anticipated performance of the testbed, particularly for when it moves into a vacuum environment. To model the surface deformation due to thermal expansion and the resultant wavefront degradation, Zemax OpticStudio was used to import finite element analysis (FEA) data directly onto the mirror surfaces. Figure 8 shows the CDEEP wavefront in the flight instrument evaluated at the FSM pupil before and after thermal deformation in a worst case scenario without payload thermal control or correction from the LLOWFS. Simulations also for the flight instrument have shown that the LLOWFS can correct for this and improves contrast by a factor of 100, and we expect even better performance from the testbed due to a more stable thermal environment on the ground.

Figure 8:

(Left) Wavefront evaluated at the FSM pupil before STOP analysis. (Right) Wavefront evaluated at the FSM pupil after thermal deformation. This worst-case example shows wavefront error without payload thermal control or the LLOWFS correction.

4.3

Jitter

The telescope pointing must be sufficiently stable that the star stays centered on the coronagraph. Setting pointing leakage to below 1 × 10⁻⁸ in the contrast budget requires ~6 nm RMS for a charge-6 VVC, which is set by the sensitivity to tilt, seen in Figure 6. This sets a CDEEP jitter requirement of ≲ 6 mas, which is easily met by the combination of spacecraft attitude control and the FSM using conventional closed loop control. Further margin is expected using predictive control¹² and will be simulated in the lab.

4.4

High Order WFS

High-order wavefront sensing and control for CDEEP will be performed using the science camera via Electric Field Conjugation (EFC).^{40, 41} A form of image plane wavefront sensing (WFS), EFC uses DM perturbations (“probes”) to solve a linear model of the coronagraph system that can be used to iteratively apply DM corrections and suppress residual starlight in the image plane until the desired contrast is reached in the region of interest. Using a single DM to control amplitude and phase will produce a one-sided dark hole as shown in Figure 9.

Figure 9:

The static wavefront error contrast for CDEEP coronagraph is 10^–8 after ~30 EFC iterations in 10% broadband light using a Charge-6 scalar vortex coronagraph and λ/10 PV optics.

CDEEP raw contrast is ~ 10⁻² less stringent than that required for ExoEarth imaging.⁴² Table 1 assigns a 10⁻⁸ static contrast floor to uncorrectable manufacturing error after EFC, consistent with past tests and simulations for typical λ/10 peak-to-valley (PV) or better surfaces.^{17, 43, 44} To validate these assumptions, we performed preliminary modeling of the CDEEP optical train using HCIPY³⁸ and PROPER.⁴⁵ The optical train was simplified to the DM, a charge-6 scalar vortex coronagraph (in the process of being updated to VVC), the Lyot stop, and the detector. CDEEP baselines two 10% bandwidth filters centered at 540-nm and 660-nm, simultaneously illuminating the science detector. The secondary blue channel provides simultaneous multi-band imaging to sense disk color and track speckle behavior over time⁴⁶ at a smaller IWA.

We simulated 50 EFC iterations for a 10% bandpass centered at 540 nm with 120 nm PV random surface errors applied. Broadband EFC was performed as in Belikov (2007),⁴⁷ where for each iteration we calculated the DM shape optimized for the central wavelength of 540 nm and accepted the contrast degradation in the rest of the bandpass. Thus for each iteration the broadband image was produced by summing together images taken at 10 wavelengths across the bandpass with the DM set to the aforementioned optimized shape. These simulations show that our assumptions are reasonable, and that the required contrast of 10⁻⁸ can be reached in ~30 iterations. We note, however, that these simulations were performed using “perfect” knowledge of the simulated image plane complex electric field provided by HCIpy, rather than estimates of the field by pairwise probing as will be done for the real system. Thus these numbers thus represent a lower bound and in the real system more iterations will likely be required to reach the required contrast.

4.5

Polarization

Mixing between polarization states leads to wavefront errors which can not be corrected by the DM.⁵¹ A polarization ray trace of optics upstream of the VVC was employed to quantify these errors. Polarization ray tracing (PRT) is a technique for calculating the polarization matrices for ray paths through optical systems. Polaris-M was built from the ground up to calculate polarization effects in optical systems. It is based on a 3x3 polarization ray-tracing calculus. Diffraction image formation of polarization aberration beams is then handled by vector extensions to diffraction theory.⁵²

Optics upstream of the vector vortex coronagraph were modeled to observe the Jones Pupil incident on the focal plane vortex mask. The resulting Jones Pupil of the first half of the system is shown in Figure 10a and 10b. The contrast contribution, calculated by fitting Zernikes to the off-diagonal terms of the Jones matrix, of the telescope and relay optics is < 10⁻⁸; thus, the testbed including the relay optics, lacking the telescope, is expected to have negligible polarization error.

(a) Jones Pupil magnitude entering the VVC before inserting polarizing elements. Magnitude shows roughly 70% of light passes through the system into the VVC. The off-diagonals values show negligible light leakage, with maximum transmission of 1% x-polarized light into y and vice-versa.

(b) Jones Pupil phase entering the VVC before inserting polarizing elements. On-diagonals show phase difference of π, while the off diagonals are close to identical.

4.6

Optics Specification and Final Design

After a period of downselect, a vendor was identified with available master parabolic mirrors which would allow OAPs with focal lengths similar enough to the flight model and with the required surface quality and Talbot lengths to be produced. The specifications of the testbed OAPs are reported in Table 4 and their fractional Talbot lengths are reported in Table 5. To further decrease polarization and alignment error, the off-axis distances (OADs) of our OAPs were slightly reduced and modeled with vacuum-compatible mounts in SolidWorks at each design iteration to ensure the mounting hardware would not vignette the coronagraph beam.

Table 4:

Mirror prescriptions for the Off-Axis Parabolas upstream of the VVC in the testbed.

Table 5:

The distances between optics upstream of the deformable mirror expressed in units of mm and fractional talbot distance z/zt.

The final optical design of the prototype testbed can be seen in schematic in Figure 11 and as a 3D rendering in Figure 12.

Figure 11:

The optical design for the CDEEP prototype testbed. The optical train begins at the source on the bottom left. The blue beams downstream of the VVC represent the starlight that is rejected by the reflective Lyot stop and used by the LLOWFS, while the orange represents the debris disk light that continues to the science cameras.

Figure 12:

3D rendering of the testbed optomechanical system as seen in schematic in Figure 11.