Polynomial Apodizers for Centrally Obscured Vortex Coronagraphs

A vortex phase mask is an azimuthal phase ramp that replaces the occulting Lyot spot in the focal plane of a coronagraph. It causes the phase of an electric field propagating through it to undergo a rotation ${e}^{{ic}\theta }$, where θ is the angular coordinate in the coordinate system whose origin is the center of the phase mask and c is the topological charge of the phase mask. The topological charge (hereafter referred to as "charge") determines the number of "rotations" the phase of the electric field undergoes per rotation around the center of the mask (see Figure 2 in Mawet et al. 2011a). A vortex phase mask in the focal plane of a coronagraph suppresses on-axis starlight by shifting it out of the radius of the original pupil in the subsequent Lyot plane. Vortices always carry even-numbered charges, since odd-numbered charges do not suppress on-axis starlight in unobscured pupils (Mawet et al. 2005).

The charge of a vortex coronagraph affects both its IWA and how stable the instrument must be kept to prevent on-axis starlight from "leaking" into the final image. In particular, charge 2 vortex coronagraphs permit small, $\sim 1\lambda /D$ IWAs, at the expense of sensitivity to low-order aberrations and finite stellar angular size (Mawet et al. 2009, 2011b). Charge 2 vortex coronagraphs are suitable for instruments designed to achieve contrasts of $\sim {10}^{-6};$ however, the sensitivity of the charge 2 vortex phase mask to low-order aberrations prevents it from being useful for obtaining larger contrasts (Mawet et al. 2010). Vortex coronagraphs with charges higher than 2, which offer lower sensitivity to low-order aberration at the expense of larger IWAs, are typically considered more likely candidates for terrestrial exoplanet characterizing instruments (Jenkins 2008; Mawet et al. 2010).

The same property that allows a vortex phase mask in the focal plane to reject on-axis starlight from an unobscured pupil is what causes the vortex coronagraph to lose starlight suppression in the presence of a central obscuration (Mawet et al. 2011c, 2013; Fogarty et al. 2014). Namely, multiplying the Airy pattern of a uniform aperture in the focal plane by ${e}^{{ic}\theta }$ causes the light through that aperture to disperse into a ring with an inner radius corresponding to the outer radius of the aperture when it propagates into the Lyot plane. The amplitude of this ring peaks at the aperture radius, and falls off as the radius increases. However, a central obscuration in the pupil serves to subtract a uniform disk from the center of the pupil, essentially creating two concentric Airy patterns in the focal plane. Therefore, the vortex coronagraph disperses some of the on-axis starlight into a ring around the central obscuration in the Lyot plane, putting it back into the aperture of the pupil. Depending on the the size of the central obscuration, the residual starlight in the aperture limits the contrast achievable by the vortex coronagraph to ${10}^{-2}\mbox{--}{10}^{-1}$ if left uncorrected (Mawet et al. 2011c).

We intend to solve the problem posed by telescope pupils with central obscurations by apodizing the entrance pupil of the vortex coronagraph. The coronagraphic setup we use is shown in Figure 1 and consists of three stages, which are labeled in the figure as Stages A, B, and C. Light from a directly on-axis source enters a pupil of radius R_P, which is obscured by a secondary mirror of radius R_S. At stage A, the pupil is apodized, which in Figure 1 is accomplished by using a grayscale apodizing filter. At stage B, the light passes through the vortex mask in the focal plane, before arriving at the inner Lyot stop at stage C, which blocks flux within a radius R_I in the Lyot plane. The final coronagraphic image is formed by focusing the beam in the Lyot plane onto a detector placed downstream of the coronagraph. Throughout this paper, we will use the polar coordinates $(r,\theta )$ when discussing the pupil and Lyot planes (stages A and C), and coordinates $(k,\theta )$ when discussing the focal plane (stage B) and final image (see Table 1 for an overview of the important variables used in this discussion).

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Table 1.  Summary of Important Variables for Deriving the Apodized Vortex Formalism

Summary of Variables
Description	Variable
Pupil/Lyot Plane Coordinates	$(r,\theta )$
Focal Plane Coordinates	$(k,\theta )$
Vortex Topological Charge	c
Secondary Mirror Radius	RS
Primary Mirror Radius	RP
Inner Lyot Stop Radius	RI
Vortex Operator for a Vortex of Charge c	Vc

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We define the "vortex operator" to be the operator mapping the electric field at the coronagraph entrance pupil to the electric field in the Lyot plane for the PAVC setup depicted in Figure 1. We will show that when the pupil plane is apodized by a polynomial apodization function ($A(r)={c}_{0}+{c}_{1}r+{c}_{2}{r}^{2}+...$ at Stage A in Figure 1), the vortex operator may be solved analytically. Using this analytical solution, we can constrain the coefficients of the polynomial apodization function to find optimal PAVCs that have no on-axis source flux in the Lyot plane using linear programming.

We proceed to derive the vortex operator for a PAVC with a vortex of charge c and a pupil apodization function $A(r)$, which we label ${V}_{c}[A(r)]$. Since the pupil radius in Figure 1 is R_P and the radius of the central obscuration is R_S, the field in the pupil due to an on-axis point source of unit flux is

$$\begin{eqnarray}&&P(r)={{\rm{\Pi }}}_{0,{R}_{P}}(r)-{{\rm{\Pi }}}_{0,{R}_{S}}(r),\end{eqnarray} \tag{ 1 }$$

where ${{\rm{\Pi }}}_{a,b}(x)$ is the unit boxcar function with lower bound a and upper bound b. Since the pupil is apodized by $A(r)$, the electric field at stage A, F_A, is

$$\begin{eqnarray}&&{F}_{A}(r)=A(r)P(r).\end{eqnarray} \tag{ 2 }$$

At stage B, the beam undergoes a Fourier transformation, and is acted on by a vortex phase mask of charge c in the focal plane. F_A is an axisymmetric function, so the Fourier Transform of F_A is equivalent to the zeroth order Hankel Transform, H₀. The field at stage B just before the vortex phase mask is therefore

$$\begin{eqnarray}&&{F}_{B}(k)={H}_{0}[{F}_{A}(r)].\end{eqnarray} \tag{ 3 }$$

After the beam passes through the vortex phase mask, the electric field is

$$\begin{eqnarray}&&{F}_{B2}(k,\theta )={F}_{B}(k){e}^{{ic}\theta }.\end{eqnarray} \tag{ 4 }$$

The beam is transformed back to the pupil plane (referred to in this case as the Lyot plane) at stage C. Just before the Lyot stop, the field at stage C is ${F}_{C}(r)={H}_{c}^{-1}[{F}_{B}(k)]$, where ${H}_{c}^{-1}$ is the inverse Hankel transform of order c. This can be obtained by rearranging the inverse Fourier transform:

$$\begin{eqnarray}{F}_{C}(r) & = & \displaystyle \frac{1}{2\pi }\displaystyle \int {F}_{B}(k){e}^{{ic}\theta }{e}^{i\sin {kr}\theta }{kdkd}{\theta }_{k}\\ & = & {\displaystyle \int }_{0}^{\infty }{F}_{B}(k)\left(\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{-\pi }^{\pi }{e}^{-i(c\tau -{kr}\sin \tau )}d\tau \right){kdk}\\ & = & {\displaystyle \int }_{0}^{\infty }{F}_{B}(k){J}_{c}({kr}){kdk}\\ & = & {H}_{c}^{-1}[{F}_{B}(k)],\end{eqnarray} \tag{ 5 }$$

where J_c is the Bessel function of order c and where we have dropped the final phase term. Putting Equations (2) through (5) together, we get the expression for the vortex operator: ${V}_{c}[A(r)]={H}_{c}^{-1}[{H}_{0}[A(r)P(r)]]$. Expanding this out, we get,

$$\begin{eqnarray}{V}_{c}[A(r)] & = & {\displaystyle \int }_{0}^{\infty }\left({\displaystyle \int }_{0}^{\infty }A(r^{\prime} )({{\rm{\Pi }}}_{0,{R}_{P}}(r^{\prime} )\right.\\ & & -{{\rm{\Pi }}}_{0,{R}_{S}}(r^{\prime} )){J}_{0}({kr}^{\prime} )r^{\prime} {dr}^{\prime} ){J}_{c}({kr}){kdk}\\ & = & {\displaystyle \int }_{0}^{\infty }\left({\displaystyle \int }_{{R}_{S}}^{{R}_{P}}A(r^{\prime} ){J}_{0}({kr}^{\prime} )r^{\prime} {dr}^{\prime} \right){J}_{c}({kr}){kdk}.\end{eqnarray} \tag{ 6 }$$

When the apodization function $A(r)$ is of the form rⁿ (where n is an integer), Equation (6) takes on a simple form for even charges c. For c = 2, we get,

$$\begin{eqnarray}{V}_{2}[{r}^{n}]=\left\{\begin{array}{ll}0, & \mathrm{if}\ r\lt {R}_{S}\\ \tfrac{-n}{n+2}{r}^{n}-\tfrac{2}{n+2}{R}_{S}^{n+2}\tfrac{1}{{r}^{2}}, & \mathrm{if}\ {R}_{S}\leqslant r\leqslant {R}_{P}\\ \tfrac{-2}{n+2}({R}_{S}^{n+2}-{R}_{P}^{n+2})\tfrac{1}{{r}^{2}}, & \mathrm{if}\ r\gt {R}_{P},\end{array}\right.\end{eqnarray} \tag{ 7 }$$

for c = 4,

$$\begin{eqnarray}{V}_{4}[{r}^{n}]=\left\{\begin{array}{ll}0, & \mathrm{if}\ r\lt {R}_{S}\\ \tfrac{(n-2){{nr}}^{n}}{(n+2)(n+4)}-\tfrac{4{R}_{S}^{n+2}}{n+2}\tfrac{1}{{r}^{2}} & \mathrm{if}\ {R}_{S}\leqslant r\leqslant {R}_{P}\\ +\tfrac{12{R}_{S}^{n+4}}{n+4}\tfrac{1}{{r}^{4}} & \\ -\tfrac{4({R}_{S}^{n+2}-{R}_{P}^{n+2})}{n+2}\tfrac{1}{{r}^{2}}+ & \mathrm{if}\ r\gt {R}_{P}\\ \tfrac{12({R}_{S}^{n+4}-{R}_{P}^{n+4})}{n+4}\tfrac{1}{{r}^{4}} & \end{array}\right.\end{eqnarray} \tag{ 8 }$$

and for c = 6,

$$\begin{eqnarray}&&{V}_{6}[{r}^{n}]\\ =\,\left\{\begin{array}{ll}0, & \mathrm{if}\ r\lt {R}_{S}\\ \tfrac{-n(n-2)(n-4)}{(n+2)(n+4)(n+6)}{r}^{n}-\tfrac{6{R}_{S}^{n+2}}{n+2}\tfrac{1}{{r}^{2}} & \mathrm{if}\ {R}_{S}\leqslant r\leqslant {R}_{P}\\ +\tfrac{48{R}_{S}^{n+4}}{n+4}\tfrac{1}{{r}^{4}}-\tfrac{60{R}_{S}^{n+6}}{n+6}\tfrac{1}{{r}^{6}} & \\ -\tfrac{6({R}_{S}^{n+2}-{R}_{P}^{n+2})}{n+2}\tfrac{1}{{r}^{2}} & \mathrm{if}\ r\gt {R}_{P}\\ +\tfrac{48({R}_{S}^{n+4}-{R}_{P}^{n+4})}{n+4}\tfrac{1}{{r}^{4}} & \\ -\tfrac{60({R}_{S}^{n+6}-{R}_{P}^{n+6})}{n+6}\tfrac{1}{{r}^{6}} & \end{array}\right.\end{eqnarray} \tag{ 9 }$$

For larger even charges, similar expressions of increasing length may be calculated. We show ${V}_{c}[{r}^{n}]$ for $c=2,4,6$ and $n=0,1,2,3$ in Figure 2. Equations (7)–(9) show that the vortex operator ${V}_{c}[A(r)]$ can be solved in closed form for any apodization function $A(r)$ that is in piecewise polynomial form (i.e., that can be decomposed into terms rⁿ that are nonzero over some range of radii). We also note that analytical solutions to Equation (6) for nonaxisymmetric pupil geometries exist and are presented in the Appendix.

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By taking advantage of the simple expressions for ${V}_{c}[{r}^{n}]$ in Equations (7)–(9), we can set up a linear program to find optimal apodizing masks that null an on-axis point source observed with an obstructed pupil. We search for apodizations of the form:

$$\begin{eqnarray}A(r)=\left\{\begin{array}{ll}{\displaystyle \sum }_{n=0}^{N}{a}_{n}{r}^{n}, & \mathrm{if}\ {R}_{S}\lt r\leqslant {R}_{I}.\\ {\displaystyle \sum }_{n=0}^{N}{a}_{n}{r}^{n}+{b}_{n}{r}^{n}, & \mathrm{if}\ {R}_{I}\lt r\leqslant {R}_{P},\end{array}\right.\end{eqnarray} \tag{ 10 }$$

where ${R}_{I}\geqslant {R}_{S}$ and where N is the order of the piecewise polynomial we wish to use to describe $A(r)$.

Using the closed-form solutions we found for ${V}_{c}[{r}^{n}]$, we can find linear constraints on the coefficients a_n and b_n of the apodization function such that in the Lyot plane (Stage C in Figure 1), the electric field is exactly zero at radii greater than R_I and less than R_P. Since the inner Lyot stop at Stage C covers the region $r\lt {R}_{I}$, an apodization function that is a solution to this linear program produces perfect cancellation of the on-axis starlight.

The apodization function in Equation (10) consists of two sets of components—"a_n" components, which are nonzero between R_S and R_P and zero elsewhere, and "b_n" components, which are nonzero between R_I and R_P and zero elsewhere. The b_n components can be thought of as seeing a central obscuration of radius R_I instead of R_S. Therefore, when these components are acted on by the vortex operator in Equations (7), (8), or (9), R_S is replaced by R_I. Defining $A(r)$ this way ensures that when we sum up the terms of $A(r)$ propagated into the Lyot plane, there exists a solution with zero electric field at ${R}_{I}\lt r\leqslant {R}_{P}$.

For the case of a charge 2 vortex, the constraints that allow ${V}_{2}[A(r)]=0$ in ${R}_{I}\lt r\leqslant {R}_{P}$ are

$$\begin{eqnarray}&&\displaystyle \sum _{n=0}^{N}\left(-\displaystyle \frac{2{R}_{S}^{n+2}}{n+2}{a}_{n}-\displaystyle \frac{2{R}_{I}^{n+2}}{n+2}{b}_{n}\right)=0,\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}&&{a}_{n}+{b}_{n}=0,\mathrm{if}\ n\gt 0.\end{eqnarray} \tag{ 11b }$$

For a charge 4 vortex, the constraints are

$$\begin{eqnarray}&&\displaystyle \sum _{n=0}^{N}\left(-\displaystyle \frac{4{R}_{S}^{n+2}}{n+2}{a}_{n}-\displaystyle \frac{4{R}_{I}^{n+2}}{n+2}{b}_{n}\right)=0,\end{eqnarray} \tag{ 12a }$$

$$\begin{eqnarray}&&\displaystyle \sum _{n=0}^{N}\left(\displaystyle \frac{12{R}_{S}^{n+4}}{n+4}{a}_{n}+\displaystyle \frac{12{R}_{I}^{n+4}}{n+4}{b}_{n}\right)=0,\end{eqnarray} \tag{ 12b }$$

$$\begin{eqnarray}&&{a}_{n}+{b}_{n}=0,\mathrm{if}\ n\ne (0,2).\end{eqnarray} \tag{ 12c }$$

For a charge 6 vortex, they are

$$\begin{eqnarray}&&\displaystyle \sum _{n=0}^{N}\left(-\displaystyle \frac{6{R}_{S}^{n+2}}{n+2}{a}_{n}-\displaystyle \frac{6{R}_{I}^{n+2}}{n+2}{b}_{n}\right)=0,\end{eqnarray} \tag{ 13a }$$

$$\begin{eqnarray}&&\displaystyle \sum _{n=0}^{N}\left(\displaystyle \frac{48{R}_{S}^{n+4}}{n+4}{a}_{n}+\displaystyle \frac{48{R}_{I}^{n+4}}{n+4}{b}_{n}\right)=0,\end{eqnarray} \tag{ 13b }$$

$$\begin{eqnarray}&&\displaystyle \sum _{n=0}^{N}\left(-\displaystyle \frac{60{R}_{S}^{n+6}}{n+6}{a}_{n}-\displaystyle \frac{60{R}_{I}^{n+6}}{n+6}{b}_{n}\right)=0,\end{eqnarray} \tag{ 13c }$$

$$\begin{eqnarray}&&{a}_{n}+{b}_{n}=0,\mathrm{if}\ n\ne (0,2,4).\end{eqnarray} \tag{ 13d }$$

It is also necessary that the solution corresponds to a physically real apodizing mask, which is guaranteed by the additional constraint

$$\begin{eqnarray}&&0\leqslant A(r)\leqslant 1.\end{eqnarray} \tag{ 14 }$$

If we wish to produce $A(r)$ using a classical apodizing mask, the constraints in either Equations (11), (12), or (13), and in Equation (14) are sufficient. However, if we wish to produce $A(r)$ using shaped mirrors, we also desire mask shapes that are smooth. We can constrain mask shapes to be continuous, using

$$\begin{eqnarray}&&\displaystyle \sum _{n=0}^{N}{b}_{n}{R}_{I}^{n}=0,\end{eqnarray} \tag{ 15 }$$

and smooth, using

$$\begin{eqnarray}&&\displaystyle \sum _{n=0}^{N}{{nb}}_{n}{R}_{I}^{n-1}=0.\end{eqnarray} \tag{ 16 }$$

It is interesting to note that since we are free to add degrees of freedom to the problem by increasing the order N of the piecewise polynomial describing $A(r)$, we can continue to add linear constraints while ensuring that the problem remains solvable. For example, we can ensure that the mask solution be C^k smooth for any order k.

Having specified the constraints on $A(r)$, it remains to choose a figure of merit (FOM) in order to calculate an optimal apodization function. The choice of FOM depends on both instrument performance goals and how the apodization function is produced. One of the key features of the PAVC is that it can be optimized according to different combinations of FOM and constraints.

For a PAVC with $A(r)$ produced with a classical apodizing filter, the FOM ought to maximize filter transmission, while the smoothness and continuity constraints can be neglected. Meanwhile, for a PAVC with shaped mirrors, an FOM must be selected that minimizes mirror curvature. In the following sections, we motivate the FOMs for PAVCs produced with either apodizing filters or with shaped mirrors, and demonstrate the results we obtain for a range of central obscuration sizes.

To summarize, the linear program that needs to be solved to find a perfectly nulling apodizing pupil mask is defined by the following.

• 1.  
  A FOM, which depends on the desired properties of the coronagraph, and by the set of constraints that ensure:
• 2.  
  Lyot plane nulling (Equations (11), (12), or (13)).
• 3.  
  Feasibility (Equation (14)).
• 4.  
  Continuity (Equation (15); optionally).
• 5.  
  Smoothness (Equation (16); optionally).

The apodization function for a PAVC incorporating an apodizing mask/filter at Stage A in Figure 1 ought to be optimized to permit as much flux through the filter as possible. Therefore, the linear FOM we use is the transmission through the mask,

$$\begin{eqnarray}&&T={\int }_{{R}_{I}}^{{R}_{P}}A(r){rdr}.\end{eqnarray} \tag{ 17 }$$

To calculate the FOM, we only integrate T between R_I and R_P since light inside R_I is blocked by the inner Lyot stop.

For a charge 2 PAVC, maximizing T is equivalent to maximizing total energy throughput (defined as the integrated energy in the Lyot plane that is transmitted through the pupil apodizing filter and inner Lyot stop; see Table 2 for an overview of how "throughput" is defined and used throughout this paper), since given the constraints in Equation (11), only the a₀ and b₀ terms in $A(r)$ do not sum to zero in the region ${R}_{I}\lt r\lt {R}_{P}$. We emphasize that T is not the same quantity as total energy throughput, which is a nonlinear function of the apodization. T in this case is

$$\begin{eqnarray}&&T=\displaystyle \frac{{a}_{0}+{b}_{0}}{2}({R}_{P}^{2}-{R}_{I}^{2}),\end{eqnarray} \tag{ 18 }$$

while the total energy throughput is

$$\begin{eqnarray}&&T.E.\,=\,\displaystyle \frac{{({a}_{0}+{b}_{0})}^{2}}{2}({R}_{P}^{2}-{R}_{I}^{2}),\end{eqnarray} \tag{ 19 }$$

so subject to the constraint that ${a}_{0}+{b}_{0}\geqslant 0$, it can clearly be seen that finding the solution to the linear program that maximizes ${a}_{0}+{b}_{0}$ will optimize both T and total energy throughput.

Table 2.  Definitions of Throughput

Definitions of Throughput for Coronagraphic Instruments Used in Sections 3–5
Name	Definition
Total Energy Throughput	The integrated energy in the Lyot plane of a PAVC that is transmitted through the pupil and Lyot stop. For a classically apodized PAVC, this quantity can be calculated by $$\begin{eqnarray}&&T.E.=\displaystyle \frac{2\pi {\int }_{{R}_{I}}^{{R}_{P}}{A}^{2}(r){rdr}}{\pi ({R}_{P}^{2}-{R}_{S}^{2})}.\end{eqnarray} \tag{ 33 }$$
Total Energy Off-axis Throughput	The total energy of an off-axis source in the Lyot plane. For a classicaly apodized PAVC, this quantity can be calculated by $$\begin{eqnarray}&&T.E.(r)=\displaystyle \frac{{\int }_{{R}_{I}}^{{R}_{P}}{\int }_{0}^{2\pi }{({V}_{c}[A(r)F(r,\theta )])}^{2}d\theta {rdr}}{\pi ({R}_{P}^{2}-{R}_{S}^{2})},\end{eqnarray} \tag{ 34 }$$ where $F(r,\theta )$ is the off-axis source flux in the telescope pupil.
Encircled Energy Throuhput	The integral of the energy contained in an aperture of diameter 0.7λ/D centered on the off-axis source position in the image plane of the PAVC. If $({K}_{x},{K}_{y})$ is the position of the off-axis source in coordinates centered on the on-axis star, then the encircled energy throughput is $$\begin{eqnarray}&&E.E.({K}_{r})=\displaystyle \frac{{\int }_{{({k}_{x}-{K}_{x})}^{2}+{({k}_{y}-{K}_{y})}^{2}\leqslant {(0.35\lambda /D)}^{2}}{I}^{2}({k}_{x},{k}_{y}){{dk}}_{x}{{dk}}_{y}}{\pi ({R}_{P}^{2}-{R}_{S}^{2})},\end{eqnarray} \tag{ 35 }$$ where ${K}_{r}\equiv \sqrt{{K}_{x}^{2}+{K}_{y}^{2}}$ and $I({k}_{x},{k}_{y})$ is the amplitude in the image plane. This quantity approximates the relative signal strength that would be obtained for an off-axis source when performing aperture photometry on the source position in the image plane.

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This is not the case for higher charges, so strictly speaking optimizing T will not allow us to find the maximum total energy throughput attainable with a classically apodized $c\gt 2$ PAVC. To find the maximum total energy throughput achievable by a charge 4 or higher PAVC for a given R_S, one would have to use a nonlinear optimizer rather than the linear programming routine we use here. For this reason, maximizing transmission instead of throughput is a commonly used method to linearize problems involving optimizing apodized coronagraph designs (Vanderbei et al. 2003; Carlotti et al. 2012; N'Diaye et al. 2015). Therefore, we use T as the FOM for charge 4 and 6 PAVCs, while bearing in mind that we may overlook solutions to Equations (12) and (13) that produce higher total energy throughputs than we report.

In order to explore the PAVC parameter space, we generated optimal smooth apodizing masks for charge 2, 4, and 6 vortices described by various orders of polynomial for different sized central obscurations. We also generated masks with noncontinuous apodization functions, where we did not impose the constraints in Equations (15) and (16). Profiles of smooth apodizing masks with apodization functions described by polynomials of the order of 20 are shown in Figure 3 for charge 2 through 6 PAVCs designed for a central obscuration with ${R}_{S}=0.3{R}_{P}$.

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The optimal total energy throughput achieved by our linear programming algorithm as a function of secondary mirror radius is shown in Figure 4. For each R_S, we chose the value of R_I that maximizes T. We find that the noncontinuous apodizations have the highest total energy throughput for each charge, and that smooth apodizations approach the throughput of the noncontinuous apodization for a given charge as the order of the smooth piecewise polynomial increases.

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Since $A(r)$ need not be continuous for a classically apodized PAVC, the optimal apodization function for charge 2 PAVC is the same as for the charge 2 RAVC. Therefore, the ideal classically apodized charge 2 vortex coronagraph is the RAVC. This is not surprising, since the Lyot plane nulling constraint for a charge 2 PAVC (Equation (11)) constrains the portion of the apodization function where $r\gt {R}_{I}$ to be flat. For higher charges, we find apodization functions with substantially higher total energy throughputs than the RAVC apodizations presented in Mawet et al. (2013). In fact, for a given central obscuration size, we find that the total energy throughput of the PAVC increases as the charge increases. This improvement stems from the fact that for higher charges, the apodization function at $r\gt {R}_{I}$ can be expressed by progressively higher orders of polynomial.

The increasing complexity of $A(r)$ at $r\gt {R}_{I}$ with increasing charge in turn explains why total energy throughput versus R_S for the charge 6 PAVC is not perfectly monotonic—maximizing T may start to noticeably underestimate total energy throughput through the apodizing mask. While $T={\int }_{{R}_{I}}^{{R}_{P}}[{a}_{0}+{b}_{0}\,+({a}_{2}+{b}_{2}){r}^{2}+({a}_{4}+{b}_{4}){r}^{4}]{rdr}$, the total energy throughput is $T.E.\,=\,{\int }_{{R}_{I}}^{{R}_{P}}{[{a}_{0}+{b}_{0}+({a}_{2}+{b}_{2}){r}^{2}+({a}_{4}+{b}_{4}){r}^{4}]}^{2}{rdr}$, which has an integrand with higher-ordered terms and three additional cross terms. Meanwhile, optimizing T for a charge 4 vortex provides a better approximation for optimizing throughput, since there are fewer terms in both the integrands for T and $T.E.$, and fewer cross terms in the integrand for throughput. For both the charge 4 and 6 PAVC, the theoretically optimal total energy throughput is slightly higher than what we report. However, for the purposes of demonstrating the characteristics of the PAVC, the approximate optimal total energy throughput curves we obtain by optimizing T suffice.

Total energy throughput as a function of both R_S and R_I is shown in Figure 5. We find that the total energy throughput as a function of either radius increases with charge. For the charge 2 PAVC, the optimal value of R_I increases monotonically with R_S, while the dependence of the optimal value of R_I on R_S is more complicated for the charge 4 and 6 cases. Indeed, the throughput surface for the charge 6 coronagraph exhibits a saddle point at around ${R}_{S}/{R}_{P}=0.3$ and ${R}_{I}/{R}_{P}=0.4$, owing to the competing effects of throughput loss due to increased Lyot stop size and gain in the apodizing mask transmission. Like the result that PAVC total energy throughput improves as charge increases, the behavior of the charge 4 and 6 throughput surfaces results from the increasing number of available degrees of freedom at $r\gt {R}_{I}$ as the charge increases. This is true whether or not the PAVC apodization is constrained to be smooth. As shown in Figure 5, forcing the apodization to be smooth affects the performance of the PAVC, but throughputs for both smooth and discontinuous apodizations are similar and depend on R_S and R_I in nearly the same way.

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Total and encircled energy throughputs for an off-axis source (see Table 2) as a function of the source's angular separation from the on-axis star are shown in Figure 6. We present curves depicting throughput versus angular separation for charge 2, 4, and 6 PAVCs, and for ${R}_{S}=0.1{R}_{P}$ and ${R}_{S}=0.3{R}_{P}$. By plotting the encircled energy throughout, we provide an estimate of the actual flux an observer would measure when performing aperture photometry on the final coronagraphic image (Ruane et al. 2016). Plotting the total energy throughput as well indicates the total amount of flux from the off-axis source present in the final coronagraphic image.

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In the charge 2 and 4 plots, the encircled energy throughput curve of the corresponding charge RAVC is shown for reference. The smooth charge 2 PAVC underperforms the charge 2 RAVC in terms of throughput, since the ideal charge 2 apodized vortex coronagraph is the RAVC. However, the situation is the reverse for the charge 4 PAVC, especially when ${R}_{S}=0.3{R}_{P}$. In that case, the smooth PAVC outperforms the off-axis throughput of the RAVC by a factor of 3. We do not show RAVC curves on our charge 6 plots, since in Mawet et al. (2013), charge $\gt 4$ RAVCs were not discussed due to low total energy throughput performance.

While going from an 0.1R_P central obscuration to a 0.3R_P central obscuration reduces the encircled energy throughput for an off-axis source by a little more than half for the charge 4 PAVC (from ∼50% to ∼22%), for the charge 6 PAVC the maximum off-axis throughput is essentially unchanged (changing from $\sim 52 \%$ to $\sim 50 \%$). However, when going from a central obscuration of 0.1R_P to 0.3R_P, the angular separation at which the encircled energy throughput reaches 20% increases from ∼3.5λ/D to ∼4.25λ/D for the charge 6 PAVC. By comparsion, the charge 4 PAVC with a central obscuration of 0.1R_P obtains a 20% encircled energy throughput at an angular separation of ∼2.0λ/D.

The charge 6 PAVC addresses the issue of throughput performance for coronagraphs designed for telescopes with large secondary mirrors. The charge 4 PAVC, meanwhile, represents a tradeoff between sensitivity to central obscuration size and IWA.

Either the charge 4 or charge 6 PAVC may provide the optimal basis for designing an instrument for an on-axis telescope. Depending on the secondary mirror size, as well as the relative importance of overall throughput performance and the minimum angular separation for an observable off-axis source, one version of the PAVC may be preferable over the other. The charge 2 RAVC may be preferable on ground-based telescopes with relatively small secondary mirrors when sensitivity to low-order aberrations is not a priority but a small IWA is.

We obtain expressions to describe the circularly symmetric mirror shapes we use to construct an apodizing shaped pupil following the derivation given in Vanderbei & Traub (2005; V05), with some modifications. A pair of mirrors is used to produce a desired apodization function—the first mirror "spreads out" the incoming point source flux, and the second mirror re-collimates the beam. We describe the shape of the first mirror with coordinates $(\rho ,\theta )$ and mirror height (relative to a flat mirror) $h(\rho ,\theta )$. The second mirror is in the pupil plane, so its coordinates are $(r,\theta )$ and its height is $\tilde{h}(r,\theta )$. We drop the θ dependence throughout, since the mirror shapes and apodization functions we wish to solve for are circularly symmetric, and note that the more general expressions may be found in V05.

To solve for the mirror heights, we use Equations (24), (29),⁴ and (34) in V05, which, dropping the θ dependence and other constants, are

$$\begin{eqnarray}&&{\partial }_{r}\tilde{h}=\rho (r)-r,\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&h(\rho )=\tilde{h}(r)+\displaystyle \frac{1}{2}{(r-\rho )}^{2},\end{eqnarray} \tag{ 21 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dr}}\rho {(r)}^{2}=2{{CA}}^{2}(r)r.\end{eqnarray} \tag{ 22 }$$

The extra coefficient C in Equation (22) normalizes the apodization function, so that energy is conserved across the two mirrors, such that

$$\begin{eqnarray}&&2\pi C{\int }_{0}^{{R}_{P}}{A}^{2}(r){rdr}=2\pi {R}_{P}^{2}.\end{eqnarray} \tag{ 23 }$$

In order to solve Equation (22), we set $A(r)=1$ for $r\lt {R}_{S}-\epsilon$, and use spline interpolation to smoothly connect $A({R}_{s})$ and $A({R}_{S}-\epsilon )$, where is an arbitrarily small separation. We note that it can be shown from the expression for $\rho (r)$ that the equation for the derivative of the second mirror is equivalent to the second-order Monge-Ampère equation for the second mirror shape presented in Pueyo et al. (2011) and Pueyo & Norman (2013).

We wish the optimize $A(r)$ to maximize the shaped mirrors' radii of curvature. Mirror shapes with sharper curvatures are both more difficult to manufacture and increase the magnitude of diffraction effects introduced by the optics, so we want to use the smoothest mirror shapes possible (Pluzhnik et al. 2006). We focus on the shape of the first mirror, which we empirically observe has the sharpest curvatures. The radius of curvature (reciprocal of the curvature, K) for this mirror is

$$\begin{eqnarray}&&\displaystyle \frac{1}{K}=\displaystyle \frac{{(1+{({\partial }_{\rho }h)}^{2})}^{3/2}}{| {\partial }_{\rho }^{2}h| }.\end{eqnarray} \tag{ 24 }$$

Finding ${\partial }_{\rho }h$ and ${\partial }_{\rho }^{2}h$ is difficult using the setup described in V05, so we solve for the setup in reverse, where an initial pupil with electric field amplitude $A(r)$ is reshaped by a series of mirrors to an outgoing beam of uniform amplitude. In this setup, $h(r)$ is the height of the second mirror, and the desired apodization function is ${A}^{-1}(r)$. Making these adjustments to Equations (20) and (21), we find that

$$\begin{eqnarray}&&\displaystyle \frac{1}{K}=\displaystyle \frac{{A}^{2}{\left(1+{\left(\tfrac{2}{A}-r\right)}^{2}\right)}^{3/2}}{2{\partial }_{r}A}.\end{eqnarray} \tag{ 25 }$$

To maximize the radius of curvature across the mirror shape, it follows that we want to maximize A and minimize ${\partial }_{r}A$. However, as seen in Figure 8, the minimum value of $A(r)$ depends on R_I, which in turn limits the total energy throughput. For a desired level of throughput, then, we can optimize for curvature by adopting ${\partial }_{r}A$ as the FOM to minimize.

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Mirror shapes for the charge 4 PAVC are shown in Figure 9 for ${R}_{S}=0.1{R}_{P}$ and ${R}_{S}=0.3{R}_{P}$. Mirrors for the charge 6 PAVC are shown in Figure 10. Since mirror shapes are normalized to conserve energy, the total energy throughput is

$$\begin{eqnarray}&&T.E.\,=\,\displaystyle \frac{\pi ({R}_{P}^{2}-{R}_{I}^{2})}{\pi ({R}_{P}^{2}-{R}_{S}^{2})}.\end{eqnarray} \tag{ 26 }$$

For the charge 4 PAVCs in Figure 9, the total energy throughput is 88% for ${R}_{s}=0.1{R}_{P}$ and 68% for ${R}_{S}=0.3{R}_{P}$. For the charge 6 PAVCs total energy throughput is 86% and 88%, respectively.

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The off-axis throughput curves for shaped-mirror PAVCs are shown in Figure 11. Total energy throughputs are shown as blue curves, and demonstrate a gain in throughput over the use of a grayscale apodizer for each combination of charge and secondary mirror radius we examined. Encircled energy throughputs are denoted by yellow curves. For each shaped-mirror PAVC, except for the charge 4 PAVC with secondary mirror radius ${R}_{S}/{R}_{P}=0.1$, the mirror shapes we calculate severely distort the off-axis point-spread function (PSF) and result in a narrow range of angular separations from the on-axis star where even a modest encircled energy throughput can be obtained. Therefore, we propose using inverse mirrors placed after the Lyot stop in Stage C in Figure 7, similar to the inverse optics proposed for PIAA mirrors (Guyon et al. 2005, 2014). Our estimate of the encircled energy throughput after the off-axis PSF is restored by inverse optics is shown by the green curve for each coronagraph design in Figure 11.

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We estimated restored off-axis PSFs by propagating the off-axis point source flux through the coronagraphic setup without a pair of shaped mirrors to reshape the pupil, and with the total energy normalized by the total energy in the final image plane of the apodized coronagraph. The restored encircled energy throughputs we present are therefore upper limits on what may be achieved with inverse optics. If we treat the encircled energy throughput of an off-axis source as the best estimate for the "usable" photometric flux, then using shaped mirrors with inverse optics represents an improvement over grayscale filters for the charge 4 PAVC by a factor of 2 and the charge 6 PAVC by about 50%.