HIGH-CONTRAST IMAGING WITH AN ARBITRARY APERTURE: ACTIVE COMPENSATION OF APERTURE DISCONTINUITIES

Because the pupil obscuration in an on-axis telescope is large, it will be very difficult to mitigate its impact with DMs with a limited stroke. Indeed, the main hindrance to high-contrast coronagraphy in on-axis telescope is the presence of the central obscuration: it often shadows much more than 10% of the aperture width while secondary supports and segment gaps cover ∼1%. We thus first focus on azimuthally symmetric coronagraphic designs in the presence of a central obscuration. This problem (without the support structures) has been addressed in previous publications by either using circularly symmetric pupil apodization (Soummer et al. 2011) or a series of phase masks (Mawet et al. 2011). Both solutions however are subject to limitations. The singularity at the center of the Optical Vector Vortex Coronagraph might be difficult to manufacture and a circular opaque spot thus lies in the central portion of the phase mask (<λ/D), which results in a degradation of the ideal contrast of such a coronagraph (Krist et al. 2011). The solutions in Soummer et al. (2011) result from an optimization seeking to maximize the off-axis throughput for a given focal plane stop diameter: the final contrast is absent from the optimization metric and is only a by-product of the chosen geometry. Higher contrasts are then obtained by increasing the size of the focal plane mask, and thus result in a loss in IWA.

Here we revisit the solution proposed by Soummer et al. (2011) in a slightly different framework. We recognize that, in the presence of wavefront errors, high contrast can only be achieved in an area of the field of view that is bounded by the spatial frequency corresponding to the DM's actuator spacing. We thus consider the design of an APLC which only aims at generating high contrast between IWA and the outer working angle (OWA). In order to do so, we rewrite coronagraphs described by Soummer et al. (2003) as an operator $\mathcal {C}$ which relates the entrance pupil P(r) to the electrical field in the final image plane. We first call $\hat{P}(\xi)$ the Hankel transform of the entrance pupil:

$$\begin{equation} \hat{P}(\xi) = \int _{D_{S}/2}^{D/2} P(r) J_0(r \xi) r dr, \end{equation} \tag{ 1 }$$

where D is the pupil diameter, D_S is the diameter of the secondary, and ξ is the coordinate at the science detector expressed in units of angular resolution (λ₀/D). λ₀ is the design wavelength of the coronagraph chosen to translate the actual physical size of the focal plane mask in units of angular resolution (often at the center of the bandwidth of interest). λ is then the wavelength at which the coronagraph is operating (e.g., the physical size of the focal plane mask remains constant as the width of the diffraction pattern changes with wavelength). For the purpose of the monochromatic designs presented herein λ = λ₀. Then the operator is given by

$$\begin{equation} \mathcal {C}\left[P(r)\right](\xi) = \hat{P}(\xi) - \frac{\lambda ^2}{\lambda _0^2} \int _{D_{S}/2}^{D/2} \hat{P}(\eta) K(\xi,\eta) \eta d \eta, \end{equation} \tag{ 2 }$$

where K(ξ, η) is the convolution kernel that captures the effect of the focal plane stop of diameter M_stop:

$$\begin{equation} K(\xi,\eta) = \int _{0}^{M_{{\rm stop}}/2} J_0(u \eta) J_0(u \xi) u du. \end{equation} \tag{ 3 }$$

An analytical closed form for this kernel can be calculated using Lommel functions. Note that this Equation (2) assumes that the Lyot stop is not undersized. Since we are interested in high-contrast regions that only span radially all the way up to a finite OWA, we seek pupil apodization of the form

$$\begin{equation} P(r) = \sum _{k = 0}^{N_{{\rm modes}}} p_k J_{Q}\left(\frac{r}{\alpha _{k}^{Q}}\right), \end{equation} \tag{ 4 }$$

where J_Q(r) denotes the Bessel function of the first kind of order Q and $\alpha _{k}^{Q}$ is the kth zero of this Bessel function. In order to devise optimal apodizations over obscured pupils, Q can be chosen to be large enough so that J_Q(r) ≪ 1 for r < D_S/2 (in practice we choose Q = 10). The $\alpha _{k}^{Q}$ corresponds to the spatial scale of oscillations in the coronagraph entrance pupil, and such a basis set yields high-contrast regions all the way to OWA ≃ N_modesλ₀/D. Since the operator in Equation (2) is linear, finding the optimal p_k can be written as the following linear programming problem:

$$\begin{eqnarray} &{\rm \max}_{\lbrace p_k \rbrace }\, [{\rm \min}_{r}(P(r))]\, \mbox{Under the constraints:} \end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray} &|\mathcal {C}\, [(P(r))]\,(\xi)| < 10^{-\sqrt{C}} \; \mbox{for}\; {\rm IWA} < \xi <{\rm OWA} \end{eqnarray} \tag{ 5b }$$

$$\begin{eqnarray} &{\rm max} _{r}(P(r))= 1 \end{eqnarray} \tag{ 5c }$$

$$\begin{eqnarray} & \left.|\frac{d}{dr} [P(r)] | < b. \right. \end{eqnarray} \tag{ 5d }$$

Our choice of cost function and constraints has been directed by the following rationale.

• (5a) We maximize the smallest value on the apodization function in an attempt to maximize throughput. The actual throughput is a quadratic function of the p_k. Maximizing it requires the solution of a nonlinear optimization problem (as described in Vanderbei et al. 2003b).
• (5b) The contrast constraint is enforced between the IWA and the OWA (<N_modes).
• (5c) The maximum of the apodization function is set to one (otherwise the p_k will be chosen to be sufficiently small so that the contrast constraint is met).
• (5d) The absolute value of the derivative across the pupil cannot be larger than a limit, denoted as b here. As the natural solutions of such problem are very oscillatory (or "bang bang"; Vanderbei et al. 2003a, 2003b), a smoothness constraint has to be enforced (see Vanderbei et al. 2007 for a similar case).

Note that the linear transfer function in Equation (2) can also be derived for other coronagraphs, with gray scale and phase image-plane masks, or for the case of undersized Lyot stops. As general coronagraphic design in obscured circular geometries is not our main purpose, we limit the scope of the paper to coronagraphs represented by Equation (2).

Typical results of the monochromatic optimization in Equation (5d), with λ = λ₀, are shown in Figure 2 for central obscurations of 10%, 20%, and 30%. In the first two cases the size of the focal plane stop is equal to 3λ/D, the IWA is 4λ/D, and the OWA is 30λ/D. As the size of the central obscuration increases, the resulting optimal apodization becomes more oscillatory and the contrast constraint has to be loosened in order for the linear programming optimizer to converge to a smoother solution. Alternatively increasing the size of the focal plane stop yields smooth apodizers with high contrast, at a cost to angular resolution (bottom panel with a central obscuration of 30%, a focal plane mask of radius 4λ/D, an IWA of 5λ/D, and an OWA of 30λ/D). These tradeoffs were described in Soummer et al. (2011); however, our linear programming approach to the design of pupil apodizations now imposes the final contrast instead of having it be a by-product of fixed central obscuration and focal plane stop. These apodizations can either be generated using a gray-scale screen (at a cost to throughput and angular resolution) or a series of two aspherical PIAA mirrors (for better throughput and angular resolution). In order not to lose generality, we present our results in Figure 2 considering the two types of practical implementations (classical apodization and PIAA apodization). In the case of a gray-scale amplitude screen, the angular resolution units are as defined in Equation (2) and the throughput is smaller than unity. In the case of PIAA apodization, the throughput is unity and the angular resolution units have been magnified by the field-independent centroid-based angular magnification defined in Pueyo et al. (2011b). We adopt this presentation for the remainder of the paper where one-dimensional PSFs will be presented with "APLC angular resolution units" in the bottom horizontal axis and "PIAAC angular resolution units" in the top horizontal axis.

Zoom In Zoom Out Reset image size

Note that this linear programming approach only optimizes the contrast for a given wavelength. However, since the solutions presented in Figure 2 feature contrasts below 10¹⁰, we choose not to focus on coronagraph chromatic optimizations. Instead, in order to account for the chromatic behavior of the coronagraph, the monochromatic simulations in Section 6 are carried out under the conservative assumption that the physical size of the focal plane stop is somewhat smaller than optimal (or that the operating wavelength of the coronagraph is slightly off, λ = 1.2λ₀). As a consequence, the raw contrast of the coronagraphs presented in Section 6 is ∼10⁻⁹. Note that this choice is not representative of all possible apodized pupil coronagraph chromatic configurations. It is merely a shortcut we use to cover the variety of cases presented in Section 6. In Section 7.2, we present a set of broadband simulations that include wavefront errors and the true coronagraphic chromaticity for a specific configuration and show that bandwidth is more likely to be limited by the spectral bandwidth of the wavefront control system than by the coronagraph. However, future studies aimed at defining the true contrast limits of a given telescope geometry will have to rely on solutions of the linear problem in Equation (5d) which has been augmented to accommodate for broadband observations. In theory, the method presented here can also be applied to asymmetric pupils. However, the optimization quickly becomes computationally intensive as the dimensionality of the linear programming increases (in particular when the smoothness constraint and the bounds on the apodization have to be enforced at all points of a two-dimensional array). This problem can be somewhat mitigated when seeking for binary apodizations, as shown in Carlotti et al. (2011), at a cost to throughput and angular resolution.

We have shown in Section 2 that by considering the design of pupil apodized coronagraphs in the presence of a circular central obscuration as a linear optimization problem, high contrast can be reached provided that the focal plane mask is large enough. In practice, the secondary support structures and the other asymmetric discontinuities in the telescope aperture (such as segment gaps) will prevent such levels of starlight suppression. We demonstrate that well-controlled DMs can circumvent the obstacle of spiders and segment gaps. In this section, we first set up our notations and review the physics of phase-to-amplitude modulation. We consider the system represented in Figure 3 where two sequential DMs are located between the telescope aperture and the entrance pupil of the coronagraph. In this configuration, the telescope aperture and the pupil apodizer are not in conjugate planes. This will have an impact on the chromaticity of the system and is discussed in Section 5. Without loss of generality, we work under the "folded" assumption illustrated in Figure 4 where the DMs are not tilted with respect to the optical axis and can be considered as lenses of index of refraction −1 (as discussed in Vanderbei & Traub 2005). In the scalar approximation the relationship between the incoming field, E_DM1(x, y), and the outgoing field, E_DM2(x₂, y₂), is given by the diffractive Huygens Integral:

$$\begin{equation} E_{{\rm DM}2}(x_2,y_2)=\frac{1}{ i \lambda Z} \int _{\mathcal {A}} E_{{\rm DM}1}(x,y) e^{i\frac{2 \pi }{\lambda }Q(x,y,x_2,y_2)} dx dy, \end{equation} \tag{ 6 }$$

where $\mathcal {A}$ corresponds to the telescope aperture and Q(x, y, x₂, y₂) stands for the optical path length between any two points at DM1 and DM2:

$$\begin{equation} Q(x,y,x_2,y_2) = h_1(x,y)+S(x,y,x_2,y_2)-h_2(x_2,y_2). \end{equation} \tag{ 7 }$$

S(x, y, x₂, y₂) is the free-space propagation between the DMs:

$$\begin{eqnarray} && S(x,y,x_2,y_2)\nonumber\\ &&\quad = \sqrt{(x-x_2)^{2}+(y-y_2)^{2}+(Z+h_1(x,y)-h_2(x_2,y_2))^{2}}, \nonumber\\ \end{eqnarray} \tag{ 8 }$$

where Z is the distance between the two DMs, h₁ and h₂ are the shapes of DM1 and DM2, respectively (as shown in Figure 4), and λ is the wavelength. We recognize that two sequential DMs act as a pupil remapping unit similar to PIAA coronagraph (Guyon 2003) whose ray optics equations were first derived by Traub & Vanderbei (2003). We briefly state the notation used to describe such an optical system as introduced in Pueyo et al. (2011a).

• 1.  
  For a given location at DM2, (x₂, y₂), the location at DM1 of the incident ray according to ray optics is given by

  $$\begin{eqnarray} & x_1(x_2,y_2) = f_1(x_2,y_2), \end{eqnarray} \tag{ 9a }$$

  $$\begin{eqnarray} & y_1(x_2,y_2) = g_1(x_2,y_2). \end{eqnarray} \tag{ 9b }$$
• 2.  
  Conversely, for a given location at DM1, (x₁, y₁), the location at DM2 of the outgoing ray according to geometric optics is given by

  $$\begin{eqnarray} & x_2(x_1,y_1) = f_2(x_1,y_1), \end{eqnarray} \tag{ 10a }$$

  $$\begin{eqnarray} & y_2(x_1,y_1) = g_2(x_1,y_1). \end{eqnarray} \tag{ 10b }$$
• 3.  
  Fermat's principle dictates the following relationships between the remapping functions and the shape of DM1:

  $$\begin{eqnarray} & \frac{\partial h_1}{\partial x} \left |_{(x_1,y_1)} = \frac{x_1 - f_2(x_1,y_1)}{Z}, \right.\end{eqnarray} \tag{ 11a }$$

  $$\begin{eqnarray} & \frac{\partial h_1}{\partial y} \left |_{(x_1,y_1)} = \frac{y_1 - g_2(x_1,y_1)}{Z}.\right. \end{eqnarray} \tag{ 11b }$$
• 4.  
  Conversely, if we choose the surface of DM2 to ensure that the outgoing on-axis wavefront is flat, we then find

  $$\begin{eqnarray} & \frac{\partial h_2}{\partial x} \left |_{(x_2,y_2)} = \frac{x_2 - f_1(x_2,y_2)}{Z},\right. \end{eqnarray} \tag{ 12a }$$

  $$\begin{eqnarray} & \frac{\partial h_2}{\partial y} \left |_{(x_2,y_2)} = \frac{y_2 - g_1(x_2,y_2)}{Z}.\right. \end{eqnarray} \tag{ 12b }$$

Zoom In Zoom Out Reset image size

In Pueyo et al. (2011a) we showed that one could approximate the propagation integral in Equation (6) by taking in a second-order Taylor expansion of Q(x, y, x₂, y₂) around the rays that trace (f₁(x₂, y₂), g₁(x₂, y₂)) to (x₂, y₂). In this case, the relationship between the fields at DM1 and DM2 is

$$\begin{eqnarray} && E_{{\rm DM}2}(x_2,y_2) = \frac{e^{\frac{2 i \pi }{\lambda } Z}}{ i \lambda Z} \nonumber\\ && \times \left\lbrace \int _{\mathcal {A}} E_{{\rm DM}1}(x,y) e^{\frac{ i \pi }{\lambda Z} \left[\frac{\partial f_2}{\partial x}(x- x_1)^{2} + 2 \frac{ \partial g_2}{\partial x}(x- x_1)(y- y_1) +\frac{\partial g_2}{\partial y}(y- y_1)^{2} \right]}dx dy \right\rbrace \left |_{(x_1,y_1)}.\right.\nonumber\\ \end{eqnarray} \tag{ 13 }$$

When the mirror's deformations are very small compared to both the wavelength and D²/Z, the net effect of the wavefront disturbance created by DM1 can be captured in E_DM1(x, y) and the surface of DM2 can be factored out of Equation (6). In this case x₁ = x₂, y₁ = y₂, $({\partial f_2}/{\partial x}) |_{x_1,y_1} = 1$, $({\partial f_2}/{\partial y}) |_{x_1,y_1}=0$, and $({\partial g_2}/{\partial y}) |_{x_1,y_1} = 1$. Then, Equation (13) reduces to

$$\begin{eqnarray} && E_{{\rm DM}2}(x_2,y_2) = \frac{e^{\frac{2 i \pi }{\lambda } (Z-h_2(x_2,y_2))}}{ i \lambda Z} \nonumber\\ &&\quad \times \int _{{\rm Aperture}} e^{\frac{2 i \pi }{\lambda }(h_1(x,y))} e^{\frac{ i \pi }{\lambda Z} ((x- x_2)^{2} +(y- y_2)^{2})}dx dy \quad \end{eqnarray} \tag{ 14 }$$

which is the Fresnel approximation. If moreover h₁(x, y) = λcos  ((2π/D)(mx + ny)), h₂(x, y) = −h₁(x, y), with ≪ 1, then the outgoing field is to first order

$$\begin{eqnarray} &&E_{{\rm DM}2}(x_2,y_2) \propto \frac{\pi \lambda Z (m^2+n^2) }{D^2}\lambda \epsilon \cos \left(\frac{2 \pi }{D}(m x_2+ n y_2)\right). \nonumber\\ \end{eqnarray} \tag{ 15 }$$

This phase-to-amplitude coupling is a well-known optical phenomenon called Talbot imaging and was introduced to the context of high-contrast imaging by Shaklan & Green (2006). In the small deformation regime, the phase on DM1 becomes an amplitude at DM2 according to the coupling in Equation (15). When two sequential DMs are controlled to cancel small amplitude errors, as in Pueyo et al. (2009), they operate in this regime. Note, however, that the coupling factor scales with wavelength (the resulting amplitude modulation is wavelength independent, but the coupling scales as λ): this formalism is thus not applicable to our case, for which we are seeking to correct large amplitude errors (secondary support structures and segments) with the DMs. In practice, when using Equation (15) in the wavefront control scheme outlined in Pueyo et al. (2009) to correct aperture discontinuities, this weak coupling results in large mirror shapes that lie beyond the range of the linear assumption made by the DM control algorithm. For this reason, methods outlined in the top panel of Figure 1 to correct for aperture discontinuities do not converge to high contrast. Because phase-to-amplitude conversion is fundamentally a very nonlinear phenomenon, these descending gradient methods (Bordé & Traub 2006; Give'on et al. 2007; Pueyo et al. 2009) are not suitable to find DM shapes that mitigate aperture discontinuities. We circumvent these numerical limitations by calculating DMs' shapes that are based on the full nonlinear problem (see the bottom panel of Figure 1).

In the general case, starting from Equation (13) and following the derivation described in Section 5, the field at DM2 can be written as follows:

$$\begin{eqnarray} E_{{\rm DM}2}(x_2,y_2) &=& \left\lbrace \sqrt{|\det \, [J]|} \int _{\mathcal {F}\mathcal {P}} \widehat{ E_{{\rm DM}1}}(\xi,\eta) e^{i 2 \pi (\xi f_1+\eta g_1)} \right.\nonumber\\ && \left.\times e^{- i\frac{\pi \lambda Z}{\det [J]}\left(\frac{\partial g_1}{\partial y} \xi ^2+ \frac{\partial f_1}{\partial x}\eta ^2 \right)} \; d\xi d \eta \right\rbrace \left |_{(x_2,y_2)},\right. \quad \end{eqnarray} \tag{ 16 }$$

where $\widehat{ E_{{\rm DM}1}}(\xi,\eta)$ is the Fourier transform of the telescope aperture and $\mathcal {F}\mathcal {P}$ stands for the Fourier plane. We call this integral the Stretched-Remapped Fresnel approximation (SR-Fresnel). Moreover, $\det [J(x_2,y_2)]$ is the determinant of the Jacobian of the change of variables that maps (x₂, y₂) to (x₁, y₁):

$$\begin{equation} \det\, [J(x_2,y_2)] = \left\lbrace \frac{\partial f_1}{\partial x}\frac{\partial g_1}{\partial y}- \left(\frac{ \partial g_1}{\partial x}\right)^2 \right\rbrace \left |_{(x_2,y_2)}\right.. \end{equation} \tag{ 17 }$$

In the ray optics approximation, λ ∼ 0, the nonlinear transfer function between the two DMs becomes:

$$\begin{eqnarray} E_{{\rm DM}2}(x_2,y_2) & = & \left\lbrace \sqrt{|\det\, [J]|} E_{{\rm DM}1}\left(f_1,g_1\right) \right\rbrace \left |_{(x_2,y_2)}\right., \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} \left[E_{{\rm DM}2}(x_2,y_2)\right]^{2} &=& \left\lbrace \det\, [J] \left[ E_{{\rm DM}1}\left(f_1,g_1\right) \right]^2 \right\rbrace \left |_{(x_2,y_2)}.\right. \quad \end{eqnarray} \tag{ 19 }$$

The square form (e.g., Equation (19)) of this transfer function can also be derived based on conservation of energy principles and is a generalization to arbitrary geometries of the equation driving the design of PIAA coronagraphs (Vanderbei & Traub 2005). A full diffractive optimization of the DM surfaces requires use of the complete transfer function shown in Equation (16). However, there do not yet exist tractable numerical method to evaluate Equation (16) efficiently enough in order for this model to be included in an optimization algorithm. Moreover, even solving the ray optics problem is extremely complicated: it requires to find the mapping function (f₁, g₁) which solves the nonlinear partial differential equation in Equation (19). Substituting for (f₁, g₁) and using Equation (12b) yields a second-order nonlinear partial differential equation in h₂. This is the problem that we set ourselves to tackle in the next section, and is the cornerstone of our ACAD. As a check, one can verify that in the small deformation regime (e.g., if h₁(x, y) = λcos  ((2π/D)(mx + ny)) and h₂(x, y) = −h₁(x, y)) Equation (19) yields the same phase-to-amplitude coupling as in Talbot imaging (Pueyo 2008). Equation (19) is a well-known optimal transport problem (Monge 1781), which has already been identified as underlying optical illumination optimizations (Glimm & Oliker 2002). While the existence and uniqueness of solutions in arbitrary dimensions have been extensively discussed in the mathematical literature (see Dacorogna & Moser 1990 for a review), there was no practical numerical solution published up until recently. In particular, to our knowledge, not even a dimensional solution for which the DM surfaces can be described using a realistic basis set has been published yet. We now introduce a method that calculates solutions to Equation (19) which can be represented by feasible DM shapes.

Ideally, we seek DM shapes that fully cancel all the discontinuities at the surface of the primary mirror and yield a uniform amplitude distribution, as shown in the top panel of Figure 5. A solution for a particular geometry with four secondary support structures has been derived by Lozi et al. (2009). It relies on reducing the dimensionality of the problem to the direction orthogonal to the spiders. It is implemented using a transmissive correcting plate that is a four-faced prism arranged such that the vertices coincide with the location of the spiders. The curvature discontinuities at the location of the spiders are responsible for the local remapping that removes the spiders in the coronagraph pupil. However, such a solution cannot be readily generalized to the case of more complex apertures, where the secondary support structures might vary in width, or in the presence of segment gaps. Moreover, it is transmissive and thus highly chromatic. Here we focus on a different class of solutions and seek to answer a different question. How well can we mitigate the effect of pupil discontinuities using DMs with smooth surfaces, a limited number of actuators (e.g., a limited maximal curvature), and a limited stroke? Under these constraints, directly solving Equation (19) (e.g., solving the forward problem illustrated in the top panel of Figure 5) is not tractable as both factors on the left-hand side of Equation (19) depend on h₂. More specifically, the implicit dependence of E_DM1(f₁(x₂, y₂), g₁(x₂, y₂)) on h₂ can only be addressed using finite elements' solvers, whose solutions might not be realistically representable using a DM. However, this can be circumvented using the reversibility of light and solving the reverse problem, where the two mirrors have been swapped. Indeed, since

$$\begin{eqnarray} &&x_2 = f_2(f_1(x_2,y_2),g_1(x_2,y_2)), \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} &&y_2 = g_2(f_1(x_2,y_2),g_1(x_2,y_2)), \end{eqnarray} \tag{ 21 }$$

then we have the following relationship between the determinants of the forward and reverse remappings:

$$\begin{eqnarray} &&1 = \left\lbrace \frac{\partial f_2}{\partial x}\frac{\partial g_2}{\partial y} - \left(\frac{ \partial g_2}{\partial x}\right)^2 \right\rbrace \left |_{(x_1,y_1)}\right. \left\lbrace \frac{\partial f_1}{\partial x}\frac{\partial g_1}{\partial y} \left(\frac{ \partial g_1}{\partial x}\right)^2 \right\rbrace \left|_{(x_2,y_2)}\right.. \nonumber\\ \end{eqnarray} \tag{ 22 }$$

We thus focus on the inverse problem, the bottom panel of Figure 5, that consists of first finding the surface of h₁ as the solution of

$$\begin{eqnarray} && \left\lbrace \left[ E_{{\rm DM}2}(f_2,g_2) \right]^2 \left[ \frac{\partial f_2}{\partial x}\frac{\partial g_2}{\partial y}- \left(\frac{ \partial g_2}{\partial x}\right)^2\right] \right\rbrace \left |_{(x_1,y_1)}\right. \nonumber \\ &&\quad = \left[ E_{{\rm DM}1}(x_1,y_1) \right]^2. \end{eqnarray} \tag{ 23 }$$

Since our goal is to obtain a pupil as uniform as possible, we seek a field at DM2 as uniform as possible:

$$\begin{eqnarray} && E_{{\rm DM}2}\,(f_2(x_1,y_1),g_2(x_1,y_1)) \nonumber \\ &&\quad = \sqrt{\int _{\mathcal {A}}E_{{\rm DM}1}(x,y)^2 dx dy} = \mbox{constant}. \end{eqnarray} \tag{ 24 }$$

Moreover, we are only interested in compensating asymmetric structures located between the secondary and the edge of the primary. We thus only seek to find (f₂, g₂) such that

$$\begin{eqnarray} && E_{{\rm DM}1}(x_1,y_1) = A(x_1,y_1) \nonumber\\ &&\quad = \left(\left[ P(x,y) - (1 - P_{O}(x,y)) \right] \times e^{- \frac{x^2+y^2}{\omega ^2}} \right) \left |_{(x_1,y_1)},\right. \quad \end{eqnarray} \tag{ 25 }$$

where P_O(x₁, y₁) is the obscured pupil, without segments or secondary supports. Finally, we focus on solutions with a high contrast only up to a finite OWA. We artificially taper the discontinuities by convolving the control term in the Monge–Ampere equation, [P(x, y) − (1 − P_O(x, y))], with a Gaussian of width ω. Note that this tapering is only applied when calculating the DM shapes via solving the reverse problem. When the resulting solutions are propagated through Equation (19) we use the true telescope pupil for E_DM1(x₁, y₁). The parameter ω has a significant impact on the final post-coronagraphic contrast. Indeed we are here working with a merit function that is based on a pupil-plane residual, while ideally our cost function should be based on image-plane intensity. By convolving the control term in the reverse Monge–Ampere equation, we low-pass filter the discontinuities. This is equivalent to giving a stronger weight to low-to-mid spatial frequencies of interest in the context of exoplanets imaging. For each case presented in Section 6, we calculate our DM shapes over a grid of values of ω which correspond to low-pass filters with cutoff frequencies ranging from ∼OWA to ∼2 OWA and we keep the shapes which yield the best contrast. This somewhat ad hoc approach can certainly be optimized for higher contrasts. However, such an optimization is beyond the scope of the present manuscript.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The problem we are seeking to solve is illustrated in the second panel of Figure 5. In this configuration, the full second-order Monge–Ampere equation can be written as

$$\begin{eqnarray} &&\left(1+Z\frac{\partial ^2 h_1}{\partial x^2}\right) \left(1+Z\frac{\partial ^2 h_1}{\partial y^2}\right) - \left(Z\frac{ \partial ^2 h_1}{\partial x \partial y}\right)^2 = A(x,y)^2, \nonumber\\ \end{eqnarray} \tag{ 26 }$$

where we have dropped the (x₁, y₁) dependence for clarity. Since we are interested in surface deformations which can realistically be created using a DM, we seek for a Fourier representation of the DMs' surface:

$$\begin{eqnarray} h_1(x,y) &=& \frac{D^2}{Z}H_1(X,Y) \nonumber \\ &=& \frac{D^2}{Z} \sum _{n=-N/2}^{N/2} \sum _{m=-N/2}^{N/2} a_{m,n} e^{i \frac{2 \pi }{D} (m X +n X)} \qquad \qquad \end{eqnarray} \tag{ 27 }$$

with $a_{-m,-n}= a_{-m,-n}^{\star }$, where N is the limited number of actuators across the DM. Note that we have normalized the dimensions in the pupil plane X = x/D, Y = y/D. The normalized second-order Monge–Ampere equation is then

$$\begin{equation} \left(1+\frac{\partial ^2 H_1}{\partial X^2}\right) \left(1+\frac{\partial ^2 H_1}{\partial Y^2}\right) - \left(\frac{ \partial ^2 H_1}{\partial X \partial Y}\right)^2 = A(X,Y)^2. \end{equation} \tag{ 28 }$$

For each configuration in this paper we first solve Equation (28) and then transform the normalized solution in physical units, which depends on the DMs' diameter D and their separation Z.

Over the past few years, a number of numerical algorithms aimed at solving Equation (28) have emerged in the literature (Loeper & Rapetti 2005; Benamou et al. 2010). Here we summarize our implementation of two of them: an explicit Newton method (Loeper & Rapetti 2005) and a semi-implicit method (Froese & Oberman 2012). We do not delve into the proof of convergence of each method, they can be found in Loeper & Rapetti (2005), Benamou et al. (2010), and Froese & Oberman (2012). Note that Zheligovsky et al. (2010) discussed both approaches in a cosmological context and devised Fourier-based solutions. Here we are interested in a two-dimensional problem and we outline below the essence of each algorithm.

4.2.1. Explicit Newton Algorithm

This method was first introduced by Loeper & Rapetti (2005) and relies on the fact that Equation (28) can be rewritten as

$$\begin{eqnarray} \det \left[ \left(\begin{array}{@{} cc @{}}1+\frac{\partial ^2 H_1}{\partial X^2} & \frac{ \partial ^2 H_1}{\partial X \partial Y} \\ \frac{ \partial ^2 H_1}{\partial X \partial Y} & 1+\frac{\partial ^2 H_1}{\partial Y^2} \end{array} \right) \right] = \det [{\rm Id}+\mathcal {H}(H_1(X,Y))], \nonumber\\ \end{eqnarray} \tag{ 29 }$$

where $\mathcal {H}(\cdot)$ is the two-dimensional Hessian of a scalar field and Id is the identity matrix. If one writes H₁ = u + v with ||v|| ≪ ||u||, then

$$\begin{eqnarray} && \det\, [{\rm Id}+\mathcal {H}(u + \delta v)] = \det\, [{\rm Id}+\mathcal {H}(u)] \nonumber\\ &&\quad +\delta \; {\rm Tr}\, [ ({\rm Id}+\mathcal {H}(u))^{\dagger ^T} \mathcal {H}(v)] +o(\delta ^2), \end{eqnarray} \tag{ 30 }$$

where $(\cdot)^{\dagger ^T}$ denotes the transpose of the comatrix. Equation (28) can thus be linearized as

$$\begin{eqnarray} && \left(1+\frac{\partial ^2 u}{\partial Y^2}\right) \frac{\partial ^2 v}{\partial X^2} + \left(1+\frac{\partial ^2 u}{\partial X^2}\right) \frac{\partial ^2 v}{\partial Y^2} - 2\frac{ \partial ^2 u}{\partial X \partial Y} \frac{ \partial ^2 v}{\partial X \partial Y} \nonumber\\ &&\quad = \left(A(X,Y)^2 - \det \left[\mathcal {H} \left(\frac{X^2+Y^2}{2}+u \right) \right] \right). \end{eqnarray} \tag{ 31 }$$

The explicit Newton algorithm relies on Equation (31) and can then be summarized as carrying out the following iterations.

• 1.  
  Choose a first guess $H_1^{0}$.
• 2.  
  At each iteration k we seek for a solution of the form $H_1^{k+1} = H_1^{k} + V^{k}$, where V^k is the DM shape update.
• 3.  
  In order to find V^k we write

  $$\begin{eqnarray} L_{E} \left(H_1^{k},V^{k} \right) &=& \left(1+\frac{\partial ^2 H_1^{k}}{\partial Y^2}\right) \frac{\partial ^2 V^{k}}{\partial X^2} + \left(1+\frac{\partial ^2 H_1^{k}}{\partial X^2}\right) \frac{\partial ^2 V^{k}}{\partial Y^2} \nonumber\\ && - 2\frac{ \partial ^2 H_1^{k}}{\partial X \partial Y} \frac{ \partial ^2 V^{k}}{\partial X \partial Y} \end{eqnarray} \tag{ 32 }$$

  $$\begin{eqnarray} R_{E} \left(H_1^{k} \right) &=& \frac{1}{\tau }\left(A^2 - \det \left[{\rm Id}+\mathcal {H} \left(H_1^{k} \right) \right] \right) \end{eqnarray} \tag{ 33 }$$

  and solve

  $$\begin{equation} L_{E} \left(H_1^{k},V^{k} \right) = R_{E} \left(H_1^{k} \right). \end{equation} \tag{ 34 }$$

  Equation (38) is a linear partial differential equation in V^k. Since we are interested in a solution which can be expanded in a Fourier series we write V^k as

  $$\begin{equation} V_1^{k}(X,Y) = \sum _{n=-N/2}^{N/2} \sum _{m=-N/2}^{N/2} v_{m,n}^{k} e^{i \frac{2 \pi }{D} (m X +n X)}. \end{equation} \tag{ 35 }$$

  Both the right-hand side and the left-hand side of Equation (38) can be written as a Fourier series, with a spatial frequency content between −N and N cycles per aperture. Equating each Fourier coefficient in these two series yields the following linear system of (2N + 1)² equations with (N + 1)² unknowns:

  $$\begin{eqnarray} && \mbox{For all} \; m_0,n_0 \; \in \; [-N, N] \mbox{: }\int _{{\rm DM}} e^{i 2 \pi (m_0 X+n_0 Y)} \nonumber\\ &&\quad \times \left[ L_{E} \left(H_1^{k},V^{k} \right) - R_{E} \left(H_1^{k} \right) \right] dX dY = 0. \end{eqnarray} \tag{ 36 }$$

  When searching for V^k as a Fourier series over the square geometry chosen here, this inverse problem is always well posed.
• 4.  
  Update the solution $H_1^{k+1} = H_1^{k} + V_1^{k+1}$.

The convergence of this algorithm relies on the introduction of a damping constant τ > 1. Loeper & Rapetti (2005) showed that as long as $({X^2+Y^2}/{2})+H_1^{k}$ remains convex, which is always true for ACAD with reasonably small aperture discontinuities, there exists a τ large enough so that this algorithm converges toward a solution of Equation (28). However, since this algorithm is gradient based, it is not guaranteed that it converges to the global minimum of the underlying nonlinear problem. In order to avoid having this solver stall in a local minimum, we follow the methodology outlined by Froese & Oberman (2012) and first carry out a series of implicit iterations to get within a reasonable neighborhood of the global minimum.

4.2.2. Implicit Algorithm

This algorithm, along with its convergence proof, is thoroughly explained in Froese & Oberman (2012). It relies on rewriting Equation (28) as

$$\begin{equation} \frac{\partial ^2 H_1}{\partial X^2}+\frac{\partial ^2 H_1}{\partial Y^2} = \sqrt{\det\, [{\rm Id}+\mathcal {H}(H_1(X,Y))]^2+2 A(X,Y)^2}. \end{equation} \tag{ 37 }$$

The implicit method consists of carrying out the following iterations.

• 1.  
  Choose a first guess $H_1^{0}$.
• 2.  
  In order to find H^{k + 1} we write

  $$\begin{eqnarray*} && L_{I} \left(H_1^{k},V^{k} \right) =\frac{\partial ^2 H_1^{k+1}}{\partial X^2}+\frac{\partial ^2 H_1^{k+1}}{\partial Y^2} \\ && R_{E} \left(H_1^{k} \right) = \sqrt{\det\, [{\rm Id}+\mathcal {H}(H_1(X,Y))]^2+2 A(X,Y)^2} \end{eqnarray*}$$

  and solve

  $$\begin{equation} L_{I} \left(H_1^{k+1} \right) = R_{I} \left(H_1^{k} \right). \end{equation} \tag{ 38 }$$

  This problem is a linear system of (N + 1)² equations with (N + 1)² unknowns and can be solved using projections on a Fourier basis:

  $$\begin{eqnarray} && \mbox{For all} \; m_0,n_0 \; \in \; [-N, N] \mbox{: }\int _{{\rm DM}} e^{i 2 \pi (m_0 X+n_0 Y)} \nonumber\\ &&\quad \times \left[ L_{I} \left(H_1^{k},V^{k} \right) - R_{I} \left(H_1^{k} \right) \right] dX \;dY = 0. \end{eqnarray} \tag{ 39 }$$

  Note that the term under the square root in $R_{I}(H_1^{k})$ is guaranteed to be positive at each iteration.
• 3.  
  Iterate over k.

The inverse problem in Equation (39) is always well posed, for any basis set or pupil geometry, while the explicit Newton method runs into convergence issues when not using a Fourier basis over a square. When seeking to use a basis set that is more adapted to the geometry of the spiders and segments or when using a trial influence function basis for the DM, the implicit method is the most promising method. In this paper, we have limited our scope to solving the reverse problem in the bottom panel of Figure 5 and used a Fourier representation for the DM. In this case we are able to use both methods. In order to make sure that the algorithm converges toward the true solution of Equation (28) we first run a few tens of iterations of the implicit method and, once it has converged, we seek for a more accurate solution using the Newton algorithm. Typical results are shown in Figure 6 where most of the residual error resides in the high spatial frequency content (e.g., above N cycles per aperture). Our solutions are limited by the non-optimality of the Fourier basis to describe the mostly radial and azimuthal structures present in telescopes' apertures. Moreover, the DM shape is the result of the minimization of a least-squares residual in the virtual end-plane of the reverse problem, with little regard to the spatial frequency content of the solution in the final image plane of the coronagraph. While this method yields significant contrast improvements, as reported in Section 6, we discuss in Section 7 how it can be refined for higher contrast.

Zoom In Zoom Out Reset image size

Once the surface of DM1 has been calculated as a solution of Equation (19), we compute the surface of DM2 based on Equation (12b), which stems from enforcing flatness of the outgoing on-axis wavefront. We seek a Fourier representation for the surface of DM2:

$$\begin{eqnarray} h_2(x,y) &=& \frac{D^2}{Z}H_2(X,Y) \nonumber \\ &=&\frac{D^2}{Z} \sum _{n=-N/2}^{N/2} \sum _{m=-N/2}^{N/2} b_{m,n} e^{i \frac{2 \pi }{D} (m X +n X)}. \qquad \qquad \end{eqnarray} \tag{ 40 }$$

Plugging the solution found in the previous step for h₁ into Equation (11b) yields a closed form for the normalized remapping functions, (F₂, G₂):

$$\begin{eqnarray*} F_2(X_1,Y_1) &=& X_1 - \sum _{n=-N/2}^{N/2} \sum _{m=-N/2}^{N/2} i 2 \pi m \; a_{m,n} \; e^{i 2 \pi (m X_1 +n Y_1)}, \\ G_2(X_1,Y_1) &=& Y_1 - \sum _{n=-N/2}^{N/2} \sum _{m=-N/2}^{N/2} i 2 \pi n \; a_{m,n} \; e^{i 2 \pi (m X_1 +n Y_1)}. \end{eqnarray*}$$

Then the normalized version of Equation (12b) can be rewritten as

$$\begin{eqnarray} L_{x}(X_1,Y_1) &=& \sum _{m=-N/2}^{N/2} i 2 \pi m\; b_{m,n} \; e^{i 2 \pi (m F_2(X_1,Y_1) +n F_2(X_1,Y_1))}, \nonumber \\ R_{x}(X_1,Y_1) &=& X_1 - F_2(X_1,Y_1), \nonumber \\ L_{x}& =& R_{x}, \end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray} L_{y}(X_1,Y_1)&=& \sum _{m=-N/2}^{N/2} i 2 \pi n\; b_{m,n} \; e^{i 2 \pi (m F_2(X_1,Y_1) +n F_2(X_1,Y_1))}, \nonumber \\ R_{y}(X_1,Y_1)&=& Y_1 - G_2(X_1,Y_1), \nonumber \\ L_{y} &=& R_{y}. \end{eqnarray} \tag{ 42 }$$

We then multiply each side of Equations (41) and (42) by

$$\begin{equation*} e^{i 2 \pi (m_0 F_2(X_1,Y_1) +n_0 F_2(X_1,Y_1))} \det \left[ {\rm Id}+ \mathcal {H}\left(H_1^{k}(X_1,Y_1)\right) \right], \nonumber \end{equation*}$$

where (m₀, n₀) corresponds to a given DM spatial frequency.

Integrating over the square area of the DM and using the orthogonality of the Fourier basis yield the following system of 2(N + 1)² equations with (N + 1)² real unknowns:

$$\begin{eqnarray*} &&\mbox{ For all } (m_0,n_0) \mbox{:} \nonumber \\ && 2 \pi i \; m_0 b_{m_0,n_0} = \int _{{\rm DM}} R_{x}(X,Y)\det \left[ {\rm Id} +\mathcal {H} \left(H_1^{k}(X,Y) \right) \right] \nonumber\\ &&\qquad \qquad \quad \, \, \times e^{i 2 \pi (m_0 F_2(X,Y) +n_0 F_2(X,Y))}\; dX dY. \nonumber \\ &&\mbox{ For all } \; (m_0,n_0) \mbox{:} \nonumber \\ && i 2 \pi i \; n_0 b_{m_0,n_0} = \int _{{\rm DM}} R_{x}(X,Y) \det\, \left[{\rm Id} + \mathcal {H}\left(H_1^{k}(X,Y)\right)\right] \nonumber\\ &&\qquad \qquad \quad \, \, \times e^{i 2 \pi (m_0 F_2(X,Y) +n_0 F_2(X,Y))}\; dX dY. \nonumber \end{eqnarray*}$$

We then find H₂, the normalized surface of DM2, by solving this system in the least-squares sense.

Once the Monge–Ampere equation has been solved, the calculation of the surface of the second mirror is a much easier problem. Indeed, by virtue of the conservation of the on-axis optical path length, finding the surface of DM2 only consists of solving a linear system (see Traub & Vanderbei 2003).

The method described above does not enforce any boundary conditions associated with Equation (28). One practical set of boundary conditions consists of forcing the edges of each DM to map to each other:

$$\begin{eqnarray} F_i\left(\pm \frac{1}{2},Y\right) &=& {\pm}\frac{1}{2}, \end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray} F_i\left(X,\pm \frac{1}{2}\right) &=& X, \end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray} G_i\left(\pm \frac{1}{2},Y\right) &=&Y, \end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray} G_i\left(X,\pm \frac{1}{2}\right) &=& {\pm}\frac{1}{2}, \end{eqnarray} \tag{ 46 }$$

with i = 1, 2. These correspond to a set of Neumann boundary conditions in H₁(X, Y) and H₂(X, Y). These boundary conditions can be enforced by augmenting the dimensionality of the linear systems on Equations (36) and (39); however, doing so increases the residual least-squares errors and thus hampers the contrast of the final solution. Moreover, Figures 7 and 8 show that, because of the one-to-one remapping near the DM edges in the control term of the reverse problem, the boundary conditions are almost met in practice. For the remainder of this paper we thus do not include boundary conditions when calculating the DM shapes when solving for H₁(X, Y) in Equation (28) since, in the worse case, only the edge rows and columns of the DMs' actuator will have to be sacrificed in order for the edges to truly map each other.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For a given pupil geometry we have calculated (H₁, H₂). We then convert the DM surfaces to real units, (h₁, h₂), by multiplication with D²/Z. We evaluate the remapping functions using Equations (11b) and (12b) and obtain the field at the entrance of the coronagraph in the ray optic approximation

$$\begin{eqnarray} && E_{{\rm DM}2}(x_2,y_2) \nonumber\\ && = \left\lbrace \sqrt{\det\, [J]} E_{{\rm DM}1}\left(f_1,g_1\right) e^{i \frac{2 \pi }{\lambda } \left(S(f_1,g_1) + h_{1}(f_1,g_1) - h_{2} \right)} \right\rbrace \left |_{(x_2,y_2)}\right., \nonumber\\ \end{eqnarray} \tag{ 47 }$$

where the exponential factor corresponds to the optical path length through the two DMs. Even if the surface of the DMs has been calculated using only a finite set of Fourier modes, we check that the optical path length is conserved.

Figure 9 shows that since the curvature of the DMs is limited by the number of modes N, our solution does not fully map out the discontinuities induced by the secondary supports and segments. However, they are significantly thinner and one can expect that their impact on contrast will be attenuated by orders of magnitude. In order to quantify the final coronagraphic contrasts of our solution, we then propagate it through an APLC coronagraph designed using the method in Section 2. In the case of a hexagon-based primary (such as JWST), we use a coronagraphic apodizer with a slightly oversized secondary obscuration and undersize outer edge in order to circularize the pupil. Note that this choice is mainly driven by the type of coronagraph we chose in Section 2 to illustrate our technique. Since the DM control strategy presented in this section is independent of the coronagraph, it can be generalized to any of the starlight suppression systems which have been discussed in the literature. For succinctness, we present our results using coronagraph solely based on using pupil apodization (either in an APLC or in a PIAAC configuration). Results for a JWST geometry are shown in Figures 11 and 12 and discussed in Section 6. Equation (47) assumes that the propagation between the two DMs occurs according to the laws of ray optics. In the next section we derive the actual diffractive field at DM2, e.g., Equation (16), and show that in the pupil remapping regime of ACAD, edge ringing due to the free-space propagation is actually smaller than in the Fresnel regime.

Zoom In Zoom Out Reset image size

ACAD is based on ray optics. It is an inherently broadband technique, and provided that the coronagraph is optimized for broadband performance ACAD will provide high contrast over large spectral windows. However, when taking into account the edge diffraction effects that are captured by the quadratic integral in Equation (16), the true propagated field at DM2 becomes wavelength dependent. More specifically, when λ is not zero then the oscillatory integral superposes on the ray optics field a series of high spatial frequency oscillations. In theory, it would be best to use this as the full transfer function to include chromatic effects in the computation of the DMs' shapes. However, as discussed in Section 4, solving the nonlinear Monge–Ampere equation is already a delicate exercise, and we thus have limited the scope of this paper to ray optics solutions. Nonetheless, once the DMs' shapes have been determined using ray optics, one should check whether or not the oscillations due to edge diffraction will hamper the contrast. This approach is reminiscent of the design of PIAA systems where the mirror shapes are calculated first using geometric optics and are then propagated through the diffractive integral in order to check a posteriori whether or not the chromatic diffractive artifacts are below the design contrast (Pluzhnik et al. 2010). In this section, we detail the derivation of Equation (16) that is the diffractive integral for the two DMs remapping system and use this formulation to discuss the diffractive properties of ACAD.

We start with the expression of the second-order diffractive field at DM2 as derived in Pueyo et al. (2011a) and Equation (13):

$$\begin{eqnarray*} && E_{{\rm DM}2}(x_2,y_2) = \frac{1}{ i \lambda Z}\left\lbrace \int E_{{\rm DM}1}(x,y)\right. \nonumber\\ &&\quad \times \left. e^{\frac{ i \pi }{\lambda Z} \left[ \frac{\partial f_2}{\partial x}(x- x_1)^{2} + 2 \frac{ \partial g_2}{\partial x}(x- x_1)(y- y_1) +\frac{\partial g_2}{\partial y}(y- y_1)^{2} \right]}dx dy \right\rbrace \left|_{(x_1,y_1)}\right..\nonumber \end{eqnarray*}$$

We write E_DM1(x, y) as its inverse Fourier transform

$$\begin{equation} E_{{\rm DM}1}(x,y) = \int \widehat{E}_{{\rm DM}1}(\xi,\eta) e^{i 2 \pi (x \xi + y \eta)} \; d \xi d \eta \end{equation} \tag{ 48 }$$

and insert this expression in Equation (13). Completing the squares in the quadratic exponential factor then yields

$$\begin{eqnarray} && E_{{\rm DM}2}(x_2,y_2) = \frac{1}{ i \lambda Z} \left\lbrace \int \widehat{E}_{{\rm DM}1}(\xi,\eta) I_1(f_1,g_1)e^{i 2 \pi [ f_1 \xi + g_1\eta ]}\right. \nonumber\\ &&\quad \times \left. e^{- i \pi \lambda Z \left(\frac{\partial f_2}{\partial y} \left|_{(f_1,g_1)} \xi ^2+ \frac{\partial g_2}{\partial x} \right|_{(f_1,g_1)} \eta ^2 \right)} \; d \xi d \eta \right\rbrace \left |_{(x_2,y_2)}\right., \end{eqnarray} \tag{ 49 }$$

with

$$\begin{eqnarray} && I_1(y_1,x_1) \nonumber\\ && = \left\lbrace \dsty\int e^{\frac{ i \pi }{\lambda Z} \left[ \frac{\partial f_2}{\partial x} \left(x- x_1 - \frac{\xi \lambda Z}{\frac{\partial f_2}{\partial x}} \right)^{2} + 2 \frac{ \partial g_2}{\partial x} (x- x_1)\,(y- y_1) +\frac{\partial g_2}{\partial y} \left(y- y_1 - \frac{\eta \lambda Z}{\frac{\partial g_2}{\partial x}} \right)^{2} \right]} dx dy \right\rbrace \left |_{(x_1,y_1)}. \right. \nonumber\\ \end{eqnarray} \tag{ 50 }$$

The integral over space, I₁(y₁, x₁), is the integral of a complex Gaussian and can be evaluated analytically. This yields

$$\begin{eqnarray} && E_{{\rm DM}2}(x_2,y_2) = \left\lbrace \frac{1}{\sqrt{ \left|\frac{\partial f_2}{\partial x} \frac{\partial g_2}{\partial y} - \left(\frac{ \partial g_2}{\partial x} \right)^2 \right|} } \int \widehat{E}_{{\rm DM}1}(\xi,\eta) \right. \nonumber\\ &&\quad \times \left. e^{i 2 \pi [ x_1 \xi + y_1\eta ]}e^{- i \pi \lambda Z \left(\frac{\partial f_2}{\partial y}\xi ^2+ \frac{\partial g_2}{\partial x} \eta ^2 \right)} d \xi d \eta \vphantom{\frac{1}{\sqrt{ \left|\frac{\partial f_2}{\partial x} \frac{\partial g_2}{\partial y} - \left(\frac{ \partial g_2}{\partial x} \right)^2 \right|} }} \right\rbrace \left|_{(x_1,y_1)}\right.. \end{eqnarray} \tag{ 51 }$$

We thus have expressed E_DM2(x₂, y₂) as a function of (x₁, y₁) = (f₁(x₂, y₂), g₁(x₂, y₂)). This expression can be further simplified: using Equations (20)–(22) one can derive $({\partial f_2}/{\partial x})|_{(x_1,y_1)} =({1}/{\det [J(x_2,y_2)]}) ({\partial g_1}/{\partial y})|_{(x_2,y_2)}$ and $({\partial g_2}/{\partial y})|_{(x_1,y_1)}=({1 }/{\det [J(x_2,y_2)]})({\partial f_1}/{\partial x})|_{(x_2,y_2)}$, which finishes to prove Equation (16):

$$\begin{eqnarray} && E_{{\rm DM}2}(x_2,y_2) = \left\lbrace \sqrt{|\det [J]|} \int _{\mathcal {F}\mathcal {P}} \widehat{ E_{{\rm DM}1}}(\xi,\eta) e^{i 2 \pi (\xi f_1+\eta g_1)}\right. \nonumber\\ &&\quad \times \left. e^{- i\frac{\pi \lambda Z}{\det [J]}\left(\frac{\partial g_1}{\partial y} \xi ^2+ \frac{\partial f_1}{\partial x}\eta ^2 \right)} \; d\xi d \eta \right\rbrace \left |_{(x_2,y_2)}\right.. \end{eqnarray} \tag{ 52 }$$

This expression is very similar to a modified Fresnel propagation and can be rewritten as such:

$$\begin{eqnarray} && E_{{\rm DM}2}(x_2,y_2) = \left\lbrace \int _{\mathcal {A}} E_{{\rm DM}1}(x,y)\right. \nonumber\\ && \quad \times \left. e^{- i\frac{\pi \det [J]}{\lambda Z} \left(\left(\frac{\partial g_1}{\partial y} \right)^{-1} (x-f_1)^2+ \left(\frac{\partial f_1}{\partial x} \right)^{-2}(y - g_1)^2 \right)} \; dx d y \right\rbrace \left|_{(x_2,y_2)}\right.. \qquad \end{eqnarray} \tag{ 53 }$$

Because of this similarity, we call this integral the Stretched-Remapped Fresnel approximation (SR-Fresnel). Indeed, in this approximation the propagation distance is stretched by $(({{\partial g_1}/{\partial y}})/{ \det [J]}, ({{\partial f_1}/{\partial x}}) /{ \det [J]})$ and the integral is centered around the remapped pupil (f₁, g₁).

The integral form provides physical insight about the behavior of the chromatic edge oscillations. If we write

$$\begin{eqnarray} &&\Gamma _{x} = \frac{ \det [J]}{\frac{\partial g_1}{\partial y}}\left |_{(x_2,y_2)}\right., \end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray} &&\Gamma _{y} = \frac{ \det [J]}{\frac{\partial f_1}{\partial x}}\left |_{(x_2,y_2)}\right., \end{eqnarray} \tag{ 55 }$$

we can identify several diffractive regimes.

• 1.  
  When Γ_x = Γ_y = 1, the E_DM2(x₂, y₂) reduces to a simple Fresnel propagation. Edge ringing can then be mitigated using classical techniques such as pre-apodization, or re-imaging into a conjugate plane using oversized relay optics.
• 2.  
  When Γ_x, Γ_y < 1, it is as if the effective propagation length through the remapping unit was increased. This magnifies the edge chromatic ringing.
• 3.  
  When Γ_x, Γ_y > 1, it is as if the effective propagation length through the remapping unit was decreased. This damps the edge chromatic ringing.
• 4.  
  When Γ_x > 1 and Γ_y < 1, e.g., at a saddle point in the DM surface, it is as if the effective propagation length through the remapping unit was decreased in one direction and increased in the other. The edge chromatic ringing can either be damped or magnified depending on the relative magnitude of Γ_x and Γ_y.

In the case of PIAA coronagraphs, the mirror shapes are such that Γ_x, Γ_y > 1 at the center of the pupil and Γ_x, Γ_y ≪ 1 at the edges of the pupil, where the discontinuities occur. As a consequence the edge oscillations are largely magnified when compared to Fresnel oscillations (see the right panel of Figure 10), and apodizing screens are necessary in order to reduce the local curvature of the mirror's shape (as also discussed in Pluzhnik et al. 2010; Pueyo et al. 2011a). In the case of ACAD, where the x-axis is chosen to be perpendicular to the discontinuity, the surface curvature is such that Γ_x > 1, Γ_y ∼ 1 at the discontinuities inside the pupil and Γ_x > 1, Γ_y ∼ 1 elsewhere. This yields damped chromatic oscillations at the remapped discontinuities and Fresnel oscillations at the edges of the pupil (see the right panel of Figure 10). Note that Figure 10 was generated using a one-dimensional toy model that assumes Equation (16) is separable, e.g., Γ_y = 1, as described in Pueyo et al. (2011b). In practice at the saddle points of the DM surfaces, near the junction of two spiders for instance, γ_x > 1, Γ_y ≲ 1, and thus our separable model does not guarantee that chromatic edge oscillations are damped for all two dimensional configurations. However, even near the saddle points ACAD provides a strong converging remapping in the direction perpendicular to the discontinuity and very little diverging remapping in the other direction. As a consequence Γ_x ≫ 1 and Γ_y is smaller than but close to one. We thus predict that even at these locations chromatic ringing will not be amplified. Even if ACAD based on pupil remapping, its diffraction properties are qualitatively very different from PIAA coronagraphs since edge ringing is not amplified beyond the Fresnel regime at the pupil edges, and is attenuated near the discontinuities. We conclude that in most cases ACAD operates in a regime where edge chromatic oscillations are not larger than classical Fresnel oscillations, and sometimes actually smaller. As a consequence, the chromaticity of this ringing can be mitigated using standard techniques developed in the Fresnel regime and we do not expect this phenomenon to be a major obstacle to ACAD broadband operations.

Zoom In Zoom Out Reset image size

We have established that the diffractive chromatic oscillations introduced by the fact that DM1 and DM2 are not located in conjugate planes is no worse than classical Fresnel ringing from the aperture edges and can be mitigated using well-known techniques which have been developed for this regime. While a quantitative tradeoff study of how to design a high-contrast instrument which minimizes such oscillations regime is beyond the scope of this paper, we briefly recall their qualitative essence to the reader.

• 1.  
  The edges of the discontinuities in the telescope aperture can be smoothed via pupil apodization before DM1. This solution is not particularly appealing as it requires the introduction of a transmissive, and thus dispersive, component in the optical train.
• 2.  
  The distance between the two DMs can be reduced. Indeed, the DMs' deformations presented herein, for 3 cm DMs separated by 1 m, are all ⩽1μm while current technologies allow for deformations of several microns. As the edge ringing scales as Z/D², chromatic oscillations will be reduced by decreasing Z. Since the DM surfaces scale as D²/Z, reducing Z will increase the DM deformations but have little impact on the feasibility of our solution as current DM technologies can reach 4 μm strokes.
• 3.  
  The coronagraphic apodizer can be placed in a plane that is conjugate to the DM1. This can be achieved by re-imaging DM2 through a system of oversized optics (the oversizing factor increases steeply when the pupil diameter decreases). By definition, there are no Fresnel edge oscillations in such a plane. Alternatively a coronagraph without pupil apodization (amplitude or phase mask in the image plane) can be used, and in this configuration it is only sufficient for the optics to be oversized.

Note that these three solutions are not mutually exclusive and that only a full diffractive analysis, which uses robust numerical propagators that have been developed based on Equation (16), can quantitatively address the tradeoffs discussed above. The development of such propagators is our next priority. In Pueyo et al. (2011b) we laid out the theoretical foundations of such a numerical tool in the case of circularly symmetric pupil remapping and this solution has been since then practically implemented, as reported by Krist et al. (2010). Generalizing this method to a tractable propagator in the case of arbitrary remapping is a yet unsolved computational problem. In the meantime we emphasize that while the spectral bandwidth of coronagraphs whose incoming amplitude has been corrected using ACAD will certainly be limited by edge diffraction effects, but these effects are no worse than Fresnel ringing and can thus be mitigated using optical designs which are now routinely used in high-contrast instruments (see Vérinaud et al. 2010 for such discussions). For the remainder of this paper, we thus assume the diffractive artifacts have been adequately mitigated and we compute our results assuming a geometric propagation between DM1 and DM2.

6.1.1. JWST

We have illustrated each step of the calculation of the DM shapes in Section 4 using a geometry similar to JWST. This configuration is somewhat a conservative illustration of an on-axis segmented telescope as it features thick secondary supports and a "small" number of segments whose gaps diffract light in regions of the image plane close to the optical axis (the first diffraction order of a six-hexagon structure is located at ∼3λ/D). In order to assess the performances of ACAD on such an observatory architecture, we chose to use a coronagraph designed around a slightly oversized secondary obscuration of diameter 0.25 D, with a focal plane mask of diameter 8λ/D, an IWA of 5λ/D, and an OWA of 30λ/D. The field at the entrance of the coronagraph after remapping by the DMs is shown in the top right panel of Figure 11. The DM surfaces, calculated assuming 64 actuators across the pupil (N = 64 in the Fourier expansion) and DMs of diameter 3 cm separated by Z = 1 m, are shown in the middle panel of Figure 11. They are well within the stroke limit of current DM technologies. The surfaces were calculated by solving the reverse problem over an even grid of 10 cutoff low-pass spatial frequencies ranging between 30 and 70 cycles per aperture for the tapering kernel ω. The value yielding the best contrast was chosen. Note that the optimal cutoff frequency depends on the spatial scale of the discontinuities, and that higher contrasts could be obtained by choosing a set of two convolution kernels for the reverse problem and finding the optimal solution using a finer grid. However, the results in the bottom row of Figure 11 are extremely promising. Figure 12 shows a contrast improvement of a factor of 100 when compared to the raw coronagraphic PSF, which is quite remarkable for an algorithm which is not based on an image-plane metric. These results illustrate that even with a very unfriendly aperture similar to JWST one can obtain contrasts as high as envisioned for upcoming Ex-AO instruments, which have been designed for much friendlier apertures. While we certainly do not advocate using such a technique on JWST, this demonstrates that ACAD is a powerful tool for coronagraphy with on-axis segmented apertures.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

6.1.2. Extremely Large Telescopes

We now discuss the case of ELTs and provide an illustration using the example of the TMT. We considered the aperture geometry shown in the top left panel of Figure 13. It consists of a pupil 37 segments across in the longest direction and a secondary of diameter ∼0.12D which is held by three main thick struts and six thin cables. As seen in the bottom left panel of Figure 13, the impact of segment gaps is minor as they diffract light beyond the OWA of the coronagraph. When using a coronagraph with a larger OWA the segment gaps will have to be taken into account, and will have to be mitigated using DMs with a larger number of actuators. In order to obtain first-order estimates of the performances of ACAD on the aperture geometry shown in the top left panel of Figure 13, we chose to use a coronagraph designed around a slightly oversized secondary obscuration of diameter 0.15 D, with a focal plane mask of diameter 6λ/D, an IWA of 4λ/D, and an OWA of 30λ/D. The field at the entrance of the coronagraph after remapping by the DMs is shown in the top right panel of Figure 13. The DM surfaces, calculated assuming 64 actuators across the pupil (N = 64 in the Fourier expansion) and DMs of diameter 3 cm separated by Z = 1 m, are shown in the middle panel of Figure 13. They are well within the stroke limit of current DM technologies. The surfaces were calculated by solving the reverse problem over an even grid of 10 cutoff low-pass spatial frequencies ranging between 30 and 70 cycles per aperture for the tapering kernel ω. The value yielding the best contrast was chosen. The final PSF is shown in the bottom right panel of Figure 13 and features a high-contrast dark hole with residual diffracted light at the location of the spiders' diffraction structures. The impact on coronagraphic contrast of secondary supports was thoroughly studied by Martinez et al. (2008). They concluded that under a 90% Strehl ratio, the contrast in most types of coronagraphs is driven by the secondary support structures to levels ranging from 10⁻⁴ to 10⁻⁵. This, in turn, leads to a final contrast after post-processing (called Differential Imaging) of ∼10⁻⁷ to 10⁻⁸. Figure 14 shows that using ACAD on an ELT pupil yields contrasts before any post-processing which are comparable to the ones obtained by Martinez et al. (2008) after Differential Imaging. This demonstrates that should two sequential DMs be integrated into a future planet-finding instrument, setting their surface deformation according to the methodology presented above will allow this instrument to deliver very high contrast in spite of the asymmetric pupil structure of such observations. Moreover, the surface of the DMs could be adjusted to mitigate for the effect of missing segments at the surface of the primary (when for instance the telescope is operating while some segments are being serviced).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

6.2.1. Constant Area Covered by the Secondary Support Structures

In the case of ELTs with a large number of small segments (when compared to the aperture size), gaps diffract light far from the optical axis (see Figure 13 for an example). The secondary support structures are then the major source of unfriendly coronagraphic diffracted light. Under the assumption that thick structures are necessary to support the heavy secondary over the very large ELT pupils, one can use the aperture area covered by the spiders as a proxy of the secondary lift constraint. We have thus explored a series of geometries for which the number of spiders increases as they get thinner while the overall area covered by the secondary support structures remains constant. In the examples shown from Figures 15 to 18, the area covered by the secondary support structures is 1.5 times greater than in the TMT geometry discussed above. In all cases we used a coronagraph with a central obscuration of 0.15 D, with a focal plane mask of diameter 6λ/D, an IWA of 4λ/D, and an OWA of 30λ/D. The surfaces were calculated by solving the reverse problem over an even grid of 10 cutoff low-pass spatial frequencies ranging between 30 and 70 cycles per aperture for the tapering kernel. The value yielding the best contrast was chosen. This exercise leads to several conclusions pertaining to the performances of ACAD with various potential ELT geometries.

Zoom In Zoom Out Reset image size

Clocking of the spiders with respect to the DM. The top two panels of Figure 15 illustrate the importance of the clocking of the spiders with respect to the DMs actuator grid (or the Fourier grid in our case). When the secondary support structures are clocked by 45° with respect to the DM actuators they are much more attenuated by ACAD, thus yielding higher contrast. This is an artifact of the Fourier basis set chosen and would be mitigated by using DMs whose actuator placement presents circular and azimuthal symmetries (Watanabe et al. 2008).

Annulus in the PSF with a large number of spiders. When the number of secondary support struts becomes very large (>20), an interesting phenomenon occurs in the raw PSF: the spiders diffract light outside an annulus of radius N_Spiders/π/D, just as spider web masks do in the case of shaped pupil coronagraphs (Vanderbei et al. 2003b). The "bump" located beyond that spatial frequency is more difficult to attenuate using the DMs (see Figure 15 for an illustration). ACAD creates small ripples at the edges of the remapped discontinuities and when too many discontinuities are in the vicinity of each other, these ripples interfere constructively and hamper the starlight extinction level yielded by ACAD.

A lot of thin spiders is more favorable than a few thick spiders. In general, decreasing the width of the spiders while increasing their number is beneficial to the contrast obtained after ACAD as illustrated on the radial averages in Figures 16 and 18. When one increases the number of spiders while decreasing their width in a classical coronagraph, the peak intensity of the diffraction pattern of one spider decreases as the squared width of the spider. The radially averaged contrast improvement without ACAD is then somewhat lesser than the square of the spider thinning factor as it is mitigated by the increasing number of spiders. When using ACAD the spiders are seen by the coronagraph as much thinner than they actually are (by a factor τ) and thus the peak intensity of their diffraction pattern is lower by a factor of $\tau \smash{^2}$. Our numerical experiments show that τ increases when the spider width decreases. As consequence, the overall contrast gain after ACAD when decreasing the width of the spiders while increasing their number is greater than in the case of a classical coronagraph. When designing ELT secondary support structures and planning to correct for them using ACAD, increasing the number of spiders to 8 or even 12 has a beneficial impact on contrast as it enables each discontinuity to become thinner and thus to be corrected to higher contrast using the DMs. The PSFs of apertures with more than 12 spiders present diffraction structures which are poorly suited for correction with square DMs. While the contrast resulting from applying ACAD to such apertures is still a decreasing function of the number of spiders, Figure 18 shows that the net contrast gain brought by the DM-based remapping is smaller than in the more gentle cases of 12 spiders. The study presented in Figures 15–18 remains to be fully optimized for each potential design of an ELT planet-finding instrument (in particular using a finer grid of cutoff spatial frequencies). It however demonstrates the flexibility of ACAD for various aperture geometries and provides a first-order rule of thumb to design telescope apertures which are friendly to direct imaging of exoplanets: "a lot of thin spiders is more favorable than a few thick spiders." In practice, the number of spiders will be limited by effects not treated in our analysis such as the mechanical rigidity, requirements on the perfection of their periodic spacing, and glancing reflections from the sides of multiple spiders. We thus advocate that, should future ELTs be built with high-contrast exoplanetary science as a main scientific driver, then such effect ought to be thoroughly analyzed. Indeed, when using ACAD to mitigate for pupil amplitude asymmetries, a large number of thin spiders are more favorable from a contrast standpoint when using ACAD to mitigate for pupil amplitude asymmetries.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

When discussing the case of JWST, we stressed the complexity associated with the optimization of ACAD in the presence of aperture discontinuities of varying width. Carrying out such an exercise would be extremely valuable to study the feasibility of the direct imaging of exo-Earth with an on-axis segmented future flagship observatory such as ATLAST (Postman et al. 2010). However, such an effort is computationally heavy and thus beyond the scope of this paper, which focuses on introducing the ACAD methodology and illustrating using key basic examples.

So far, none of the examples in this manuscript demonstrate that ACAD can yield corrections all the way down to the theoretical contrast floor that is set by the coronagraph design. When seeking to image exo-earths from space, future missions will need to reach this limit. In order to explore this regime, we conducted a detailed study of a hypothetical on-axis monolithic telescope with four secondary support struts. To establish the true contrast limits, we varied the thickness of the spiders for each geometry. The surfaces were calculated by solving the reverse problem over an even grid of 70 cutoff low-pass spatial frequencies ranging between 30 and 70 cycles per aperture for the tapering kernel. The value yielding the best contrast was chosen. In all cases we used a coronagraph with a central obscuration of 0.15 D, with a focal plane mask of diameter 6λ/D, an IWA of 4λ/D, and an OWA of 30λ/D. Note that when using coronagraphs relying on pupil apodization these results can be readily generalized to larger circular secondary obscurations, at a loss in IWA (as shown in Figure 2). Moreover, we clocked the telescope aperture by 45° with respect to the grid of Fourier modes. We found that, indeed, the theoretical contrast floor set by the coronagraph design is met for thin spiders (0.02 D), and it is very close to be met for spiders only half the thickness of the ones currently equipping the Hubble Space Telescope (HST; 0.05 D); see Figures 19 and 20. Even in the case of thick struts (0.1 D) we find contrasts an order of magnitude higher than in the similar configuration in the top panel of Figure 16, due to our thorough optimization of the cutoff frequency of the tapering kernel and careful clocking of the aperture with respect to the actuators. On-axis telescopes are thus a viable option to image Earth analogs from space: their secondary support structures can be corrected down to contrast levels comparable to the target contrast of recent missions' concept studies (Guyon et al. 2008; Trauger et al. 2010). Since the baseline wavefront control architecture for future space coronagraphs relies on two sequential DMs, ACAD does not add any extra complexity to such missions and merely consists of controlling the DMs in order to optimally compensate for the effects of asymmetric aperture discontinuities.

Because ACAD relies on deforming the DMs' surfaces in an aspherical fashion, off-axis wavefronts seen through the two DMs' apparatus will be distorted, just as in a PIAA coronagraph (Martinache et al. 2006). However, the asphericity of the surfaces in the case of ACAD operating on reasonably thin discontinuities is much smaller than in a PIAA remapping unit. Figure 21 shows the impact on off-axis PSFs of such a distortion in the worse-case scenario of a geometry similar to JWST. We demonstrate that most of the flux remains in the central disk of radius λ/D for all sources in the field of view of the coronagraphs considered here (all the way to 30 cycles per aperture). We conclude that, because of the small deformations of the DMs, PSF distortion will not be a major hindrance in exoplanet imaging instruments whose DMs are controlled in order to mitigate for discontinuities in the aperture.

7.2.1. General Equations in the Presence of Incident Phase Errors and Discontinuities

So far we have treated primary mirrors' segmentation as a pure amplitude effect. In reality, the contrast floor in segmented telescopes will be driven by both phase and amplitude discontinuities: here we explore the impact of phase errors and discontinuities occurring before two DMs whose surfaces have been set using ACAD. There are two main phenomena to be considered. The first is the conversion of the incident wavefront phase before DM1: (2π/λ)Δh₁ into amplitude at the second mirror. The second is the projection of this wavefront phase into remapped phase errors at DM2: (2π/λ)Δh₁(f₁(x₁, y₂), g₁(x₁, y₂)). Since the remapping unit is designed using DMs, both DM1 and DM2, a complete correction could be attained in principle. However, the DMs are continuous while Δh₁ presented discontinuities. Thus, complete corrections for segmented mirrors might not be achieved in practice. Below we discuss the following two main points. (1) Even if the phase wavefront error Δh₁ has discontinuities, the phase errors within segments still drive the phase-to-amplitude conversion and thus the propagated amplitude at DM2. In that case, treatments of these phenomena that have already been discussed in the literature for monolithic apertures are still valid for small enough phase errors (Equation (56)) and smooth enough remapping functions. For ACAD remapping this smoothness constraint is naturally enforced by the limited number of actuators across the DM surface. In this case phase-to-amplitude conversion can, in principle, be corrected using DM1. (2) Remapped phase discontinuities can be corrected for a finite number of spatial frequencies using a continuous phase sheet DM. We illustrate this partial correction over a 20% bandwidth using numerical simulations of a post-ACAD half dark hole created by superposing a small perturbation, computed using a linear wavefront control algorithm, to the ACAD DM2 surface.

If the incoming wavefront is written as Δh₁ and the solution of the Monge–Ampere equation for DM1 as $h_1^{0}$, then one can conduct the analysis in Equations (6)–(12b) using $\tilde{h_1} = h_1^{0} + \Delta h_1$. Under the assumption that the surface of DM2 is set as $\tilde{h_2}$ in order to conserve optical path length, then one can rewrite the remapping as $(\tilde{f_1},\tilde{g_1})$ defined by

$$\begin{eqnarray} &&\frac{\partial \tilde{h_1}}{\partial x}\left |_{(\tilde{f_1}(x_2,y_2),\tilde{g_1}(x_2,y_2))} = \frac{\tilde{f_1}(x_2,y_2) -x_2}{Z},\right. \nonumber \\ &&\frac{\partial \tilde{h_1}}{\partial y} \left |_{(\tilde{f_1}(x_2,y_2),\tilde{g_1}(x_2,y_2))} = \frac{\tilde{g_1}(x_2,y_2) - y_2}{Z}.\right. \end{eqnarray} \tag{ 56 }$$

Moreover, if edge ringing has been properly mitigated then the ray optics solution is valid and the field at DM2 can be written as

$$\begin{eqnarray} E_{{\rm DM}2}(x_2,y_2) &=& \left\lbrace \left(\frac{ E_{{\rm DM}1}} {\left(1+\frac{\partial ^2 \tilde{h_1}}{\partial x^2} \right) \left(1+\frac{\partial ^2 \tilde{h_1}}{\partial y^2} \right) - \left(\frac{\partial ^2 \tilde{h_1}}{\partial x \partial x} \right)^2} \right)\right. \left|_{(\tilde{f_1},\tilde{g_1})}\right.\nonumber\\ && \times \left. e^{i \frac{2 \pi }{\lambda } (S(\tilde{f_1},\tilde{g_1}) + \tilde{h_{1}}(\tilde{f_1},\tilde{g_1}) -\tilde{ h_{2}})}\right\rbrace \left|_{(x_2,y_2)}\right.. \end{eqnarray} \tag{ 57 }$$

7.2.2. Impact on the Amplitude After ACAD

We first consider the amplitude profile in Equation (57): it is composed of two factors—the remapped telescope aperture, $E_{{\rm DM}1}(\tilde{f_1},\tilde{g_1})$, and the determinant of ${\rm Id} + \mathcal {H}[\tilde{h_1}]$.

The first condition necessary for the incoming wavefront not to perturb the ACAD solution is: Δh₁ is such that the remapping is not modified at the pupil locations where the telescope aperture is not zero E_DM1 ≠ 0. This results in the conditions

$$\begin{eqnarray*} \frac{\partial \Delta h_1}{\partial x} &\ll & \frac{\partial \Delta h_1^{0}}{\partial x} \mbox{ for } (x,y) \mbox{ such that } E_{{\rm DM}1}(x,y)\ne 0, \nonumber \\ \frac{\partial \Delta h_1}{\partial y} &\ll & \frac{\partial \Delta h_1^{0}}{\partial y} \mbox{ for } (x,y) \mbox{ such that } E_{{\rm DM}1}(x,y)\ne 0. \nonumber \end{eqnarray*}$$

At the locations where E_DM1 = 0 there is no light illuminating the discontinuous wavefront and thus the large local slopes at these locations have no impact on the remapping functions (f₁, g₁). These conditions are not true in segmented telescopes that are not properly phased, for which the tip-tilt error over each segment can reach several waves. However, under the assumption that the primary has been properly phased (for instance, the residual rms wavefront after phasing is expected to be ∼1/10th of a wave, similar to values expected for JWST NIRCAM), these conditions are true within the boundaries of each segment. Moreover, while the local wavefront slopes at the segment's discontinuities do not respect this condition the incident amplitude at these points is E_DM1(x, y) = 0 and they thus do not perturb the ACAD remapping solution.

The second necessary condition resides in the fact that the determinant of ${\rm Id} + \mathcal {H}(\tilde{h_1})$ is not equal to $\det [{\rm Id} + \mathcal {H}(h_1^{0})]$ at the pupil locations where the telescope aperture is not zero ought not have a severe impact on contrast. One can use the linearization in Loeper & Rapetti (2005) to show that

$$\begin{eqnarray} && \frac{1}{\det\, [{\rm Id} + \mathcal {H}(\tilde{h_1})]} \nonumber\\ &&= \frac{1}{\det\, \left[{\rm Id} \,{+}\, \mathcal {H} \left(h_1^{0} \right) \right] \left(\! 1\,{+}\, \frac{ \left(1+h_{1xx}^{0} \right) \Delta h_{1yy}+ \left(1+h_{1yy}^{0} \right) \Delta h_{1xx}- 2*h_{1xy}^{0}\Delta h_{1xy}}{\det \left[{\rm Id} + \mathcal {H} \left(h_1^{0} \right) \right]} \right)}, \nonumber\\ \end{eqnarray} \tag{ 58 }$$

$$\begin{eqnarray} \frac{1}{\det\, [{\rm Id} + \mathcal {H}(\tilde{h_1})]} &=& \frac{1}{\det \left[{\rm Id} + \mathcal {H} \left(h_1^{0} \right) \right](1+\Delta A(\Delta h_{1}))}. \end{eqnarray} \tag{ 59 }$$

The perturbation term ΔA(Δh₁) corresponds to the full nonlinear expression of the phase-to-amplitude conversion of wavefront errors that occurs in pupil remapping units. In Pueyo et al. (2011b) we derived a similar expression in the linear case, when Δh₁ ≪ λ and showed that in the pupil regions where the beam is converging this phase-to-amplitude conversion was enhanced when compared to the case of a Fresnel propagation. In a recent study, Krist et al. (2011) presented simulations predicting that this effect was quite severe in PIAA coronagraphs and can limit the broadband contrast after wavefront control unless DMs were placed before the remapping unit. In principle, ACAD will not suffer from this limitation as the first aspherical surface of the remapping unit is actually a DM that can actually compensate for Δh₁, before any phase-to-amplitude wavefront modulation occurs. Devising a wavefront controller that relies on DM1 moreover requires a computationally efficient model to propagate arbitrary wavefronts through ACAD. Such a tool was developed in Krist et al. (2010) assuming azimuthally symmetric geometries. Since devising such a tool in ACAD's case, in the asymmetric case, represents a substantial effort well beyond the problem of prescribing ACAD DM shapes, we chose not to include such simulations in the present manuscript. Since we are using DMs with a limited number of actuators, ACAD remapping is in general less severe than in the case of PIAA. We thus expect the results regarding the wavefront correction before the remapping unit reported in Krist et al. (2011) to hold. This is provided that the DM actuators can adequately capture the high spatial frequency content of Δh₁ to create a dark hole in the coronagraphic PSF. We tackle this particular aspect next when discussing the case of phase errors, in the absence of wavefront phase-to-amplitude conversion. Once again, note that while the local wavefront curvatures are very large at the segment's discontinuities, the incident amplitude at these points is E_DM1(x, y) = 0 and thus they do not have an impact on the ACAD phase-to-amplitude modulation. In practice, if the DM is not exactly located at a location conjugate to the telescope pupil, the actual wavefront discontinuities will be slightly illuminated and might perturb the remapping functions and the phase-to-amplitude conversion. While this might tighten requirements regarding the positioning of DM1 in the direction of the optical axis, we do not expect this effect to be a major obstacle to successful ACAD implementations.

7.2.3. Impact on the Phase After ACAD

In practice, when Δh₁ presents discontinuities, the surface of DM2 cannot be set to the deformation $\tilde{h_2}$ that conserves optical path length, since we work under the assumption that the DMs have a continuous phase sheet. While this has no impact on the discussion above regarding the amplitude of E_DM2, since DM1 is solely responsible for this part, it ought to be taken into account when discussing the phase at DM2. Under the assumption that Δh₁ does not perturb the nominal ACAD remapping function, one can show that the phase at DM2 is

$$\begin{eqnarray} {\rm arg}[E_{{\rm DM}2} (x_2,y_2)] &=& \frac{2 \pi }{\lambda } \left(\Delta h_1 \right(f_1^{0}(x_2,y_2),g_1^{0}(x_2,y_2) \left) \right. \nonumber\\ &&\left. + \Delta h_2 (x_2,y_2) \right), \end{eqnarray} \tag{ 60 }$$

where Δh₂(x₂, y₂) is a small continuous surface deformation superposed to the ACAD shape of DM2 and $\Delta h_1(f_1^{0}(x_2,y_2),g_1^{0}(x_2,y_2))$ is the telescope optical path length seen through the DM-based remapping unit. This second term presents phase discontinuities whose spatial scale has been contracted by ACAD. When these discontinuities are very small, their high spatial frequency content does not disrupt the ability of DM2 to correct for low-to-mid spatial frequency wavefront errors. However, as the discontinuities become larger, their high spatial frequency content can fold into the region of the PSF that the DMs seek to cancel. These "frequency folding" speckles are highly chromatic (Give'on et al. 2006) and can have a severe impact on the spectral bandwidth of a coronagraph whose wavefront is corrected using a continuous DM.

In order to assess the impact of this phenomenon, we conducted a series of simulations based on a single DM wavefront control algorithm that seeks to create a dark hole in one-half of the image plane as in Bordé & Traub (2006). We use the example of a geometry similar to JWST and work under the assumption that the discontinuous wavefront incident to the coronagraph has the same spatial frequency content as a JWST NIRCAM optical path difference that has been adjusted to 70 nm rms in order to mimic a visible Strehl similar to the near-infrared Strehl of JWST. The nonlinear wavefront and sensing and control problem associated with phasing a primary mirror to such level of precision is undoubtedly a colossal endeavor and is well beyond the scope of this paper. In this section we work under the assumption that the primary mirror either has been phased to such a level that the wavefront discontinuities are no larger than 200 nm peak to valley or that the wavefront has been otherwise corrected down to this specification using a segmented DM that is conjugate with the primary mirror. Moreover, we assume that (1) the residual post-phasing wavefront map has been characterized and can be used in order to build the linear model underlying the wavefront controller and (2) the focal plane wavefront estimator (carried out using DM diversity as in Bordé & Traub 2006 for instance) is capable of yielding an exact estimate of the complex electrical field at the science camera. Underlying this last assumption is the overly optimistic premise that wavefront will remain unchanged over the course of each high-contrast exposure. While this is not a realistic assumption, one could envision the introduction of specific wavefront sensing schemes, with architectures similar to the one currently considered for low-order wavefront sensors on monolithic apertures (Guyon et al. 2009; Wallace et al. 2011), or using a separate metrology system. The results presented here are thus limited to configurations for which segment phasing will be dynamically compensated using specific sensing and control beyond the scope of this paper. As this section merely seeks to address the controllability of wavefront errors in segmented telescopes, we chose to conduct our simulations with a perfect estimator. Finally, we use the stroke minimization wavefront control algorithm presented in Pueyo et al. (2009) to ensure convergence for as many iterations as possible. We first tested the case of a segmented telescope in the absence of ACAD, using an azimuthally symmetric coronagraph and a single DM. We sough to create a dark hole between 5 and 28 λ₀/D under a 20% bandwidth with λ₀ = 700 nm. Figure 22 shows the results of such a simulation. The DM can indeed correct for the discontinuities over a broadband in one-half of the image plane. However, the wavefront control algorithm seeks to compensate for the diffractive artifacts associated with the secondary support structures: it attenuates them on the right side of the PSF while it strengthens them on the left side of the PSF. As a result, the DM surface becomes too large at the pupil spider's location and the quasi-linear wavefront control algorithm eventually diverges for contrasts ∼10⁶. We then proceeded to simulate the same configuration in the presence of two DMs whose surface at rest was calculated using ACAD (see Figure 23). Since there does not exist a model yet to propagate arbitrary wavefronts through ACAD (the models in Krist et al. 2011 only operate under the assumption of an azimuthally symmetric remapping), we can only use the second DM for wavefront control. We work under the assumption that the incident wavefront does not perturb the nominal ACAD remapping (which is true in the case of the surface map we chose for our example) and that the arguments in Krist et al. (2010) hold so that phase-to-amplitude conversion in ACAD can be compensated by actuating DM1. Frequency folding will then be the phenomenon responsible for the true contrast limit. In this section, we are interested in exploring how this impacts the controllability of wavefront discontinuities using continuous phase-sheet DMs. We used an azimuthally symmetric coronagraph and superposed our wavefront control solution to DM2. We sought to create a dark hole between 5 and 28 λ₀/D under a 20% bandwidth with λ₀ = 700 nm. Figure 22 shows the results of such a simulation. When the incident wavefront is small enough, it is indeed possible to superpose a "classical linear wavefront control" solution to the nonlinear ACAD DM shapes in order to carve PSF dark holes. The wavefront control algorithm now yields a DM surface that does not feature prominent deformations at the location of the spiders. Most of the DM stroke is located at the edge of the segments, at the location of the wavefront discontinuities, and seek to correct the frequency folding terms associated with such discontinuities. At these locations, the DM surface eventually becomes too large and the linear wavefront control algorithm diverges. However, this divergence occurs at contrast levels much higher than when the ACAD solution is not applied to the DMs. These simulations show that indeed discontinuous phases can be corrected using the second DM of an ACAD whose surfaces have preliminarily been set to mitigate the effects of spiders and segment gaps.

Assuming that edge ringing has been properly mitigated so that the ray optics approximation underlying the calculation of the DMs' shapes is valid, one can wonder about the ultimate contrast limitations of the results presented in this manuscript. Increasing the number of actuators would have dramatic effects on contrast if the actuator count would be such that N > D/d, where d is the scale of the aperture discontinuities. Unfortunately, current DM technologies are currently far from such a requirement and the solutions presented here are in the regime where N ≪ D/d. In this regime, N only has a marginal influence on contrast when compared to the impact of the cutoff frequency of the tapering kernel. In the regime described here, varying the actuator count only changes the size of the corrected region.

The residual PSF artifacts in Figures 11–20 follow the direction of the initial diffraction pattern associated with secondary support structures and segments. When addressing the problem of aperture discontinuities by solving the Monge–Ampere equation, ACAD calculates the DM shapes based on a pupil-plane metric and thus mostly focuses on attenuating these structures with little regard to the final contrast. It is actually quite remarkable that such a pupil-only approach yields levels of starlight extinction of two to three orders of magnitude. A more appropriate metric would be the final intensity distribution in the post-coronagraphic image plane. However, as discussed in Section 3 classical wavefront control algorithms based on a linearization of the DMs' deformations around local equilibrium shapes (such as the ones presented in Bordé & Traub 2006 and Give'on et al. 2007 in the one DM case or Pueyo et al. 2009 for one or two DMs) cannot be used to compensate the full aperture discontinuities. This is illustrated in Figure 22, where the DM surface in the vicinity of spiders becomes too large after a certain number of iterations, which leads the iterative algorithm to diverge. When attempting to circumvent this problem by recomputing the linearization at each iteration, we managed to somewhat stabilize the problem for a few iterations and reached marginal contrast improvements, but the overall algorithm remained unstable unless a prohibitively small step size was used. This is the problem which motivated our effort to calculate the DM shapes as the full nonlinear solution of the Monge–Ampere equation. While doing so yields significant contrast improvements in both the case of JWST-like geometries, TMT, and similar on-axis monolithic apertures; this approach does not give a proper weight to the spatial frequencies of interest for high-contrast imaging. We mitigated this effect by giving a strong weight to the spatial frequencies of interest (in the dark hole) when solving the Monge–Ampere equation.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The next natural step is thus to use nonlinear solutions presented herein to correct for the bulk of the aperture discontinuities and to serve as a starting point for classical linearized waveform control algorithms, as illustrated in Figure 25. Figure 24 indeed illustrates that when superposing an image-plane-based wavefront controller to the Monge–Ampere ACAD solution, the contrast can be improved beyond the floor shown in Figure 12. However, one-DM solutions are of limited interest as they only operate efficiently over a finite bandwidth and over half of the image plane. ACAD yields a true broadband solution, and consequently it would be preferable to use the two DMs in the quasi-linear regime to quantify the true contrast limits of ACAD. In such a scheme, the DM surfaces are first evaluated as the solution of the Monge–Ampere equation and then adjusted using the image-plane-based wavefront control algorithm presented in Pueyo et al. (2009). However, such an exercise requires efficient and robust numerical algorithms to evaluate Equation (16). Such a tool only exists so far in the case of azimuthally symmetric remapping units (Krist et al. 2010). Developing such numerical tools is thus of primary interest to both quantifying the chromaticity and the true contrast limits achievable with on-axis and/or segmented telescopes.

Zoom In Zoom Out Reset image size