2.1.

Matrix Formalism to Calculate the Average Dark Hole Contrast

The goal of PASTIS is to model coronagraphic images in the presence of optical aberrations on a segmented primary mirror, which can be represented for example using localized Zernike polynomials. This basis is an obvious possible choice since segment-level piston, tip/tilt, focus, and astigmatism are naturally occurring aberrations from segment misalignments, for example in three-mirror anastigmat (TMA) designs such as JWST^12,58 or LUVOIR.^4,49 Although beyond the scope of this paper, the PASTIS approach can also be applied directly to any other function basis, for example to represent mirror WFEs induced by thermo-mechanical effects (mounting and backplane deformations).^59–61 In this paper, we simply expand the phase aberration in the segmented pupil ${\varphi }_s$ as a sum of local (segment-level) Zernike polynomials [Eq. (9) in Ref. 54]:

Fig. 2

Piston pair aberrations on a segmented pupil (top) and the resulting image plane intensity distributions in the DH (bottom), using the narrow-angle APLC for LUVOIR-A in an E2E propagation model. The left three panels show different interference pairs with corresponding Young-like interference fringes, while the right panel shows a random distribution of local piston on all segments of the pupil and the resulting image plane intensity. All plots appear on the same scale.

In high-contrast coronagraphy, the best contrast is not typically obtained for the perfect aperture without any aberration, but more commonly in the presence of a wavefront control solution using deformable mirrors.^30,46 Therefore, we are studying the response of the coronagraphic system to a perturbation around that solution. Defining ${\varphi }_{DH}$ as the phase solution for best DH contrast and ${\varphi }_s$ as the segmented perturbation, the phase can be divided into

High-contrast coronagraphy requires exquisite wavefront quality around the DH solution, and therefore, we assume the small aberration regime for ${\varphi }_s$, where the electric field $E(\mathbf{r})$ is well approximated as an affine function of the phase: $E(\mathbf{r})=P(\mathbf{r})e^{i\varphi (\mathbf{r})}\simeq P^{\prime }(\mathbf{r})+i{\varphi }_s(\mathbf{r})$. The phase ${\varphi }_s(\mathbf{r})$ is zero where the pupil aperture $P(\mathbf{r})$ is zero, and $P^{\prime }(\mathbf{r})$ is a complex pupil that includes the wavefront solution to produce the static DH (with both phase and amplitude contributions, and including static errors). It is noteworthy that the phase ${\varphi }_{DH}$ is not necessarily small.^30,31

Using Fourier optics for a scalar description of the electric field and of its propagation, the coronagraph propagation can be represented by a linear operator $\mathcal{C}$. This is a valid assumption for Lyot-style coronagraphs, for example an APLC^56,63,64 such as the one illustrated in Fig. 1 for the LUVOIR-A coronagraph design, or a vortex coronagraph.^65,66 High-order vortex designs would need a special treatment for the specific low-order modes (e.g., defocus) they reject perfectly,²⁷ but the tolerancing of such global modes is not the main purpose of PASTIS anyway. We can hence express the intensity distribution in the final image plane as

In most cases of interest, we will be working in a symmetrical DH. It can be shown that the spatial average of the linear cross-term in Eq. (4) over a symmetrical DH is zero [Appendix A in Ref. 54]. This simplifies Eq. (4) to a quadratic function of the phase:

The main metric used in this paper is the spatial average contrast over the extent of the DH, ${⟨\dots ⟩}_{DH}$, so using $c_0^{\prime }={⟨{|\mathcal{C}\{P^{\prime }\}|}^2⟩}_{DH}$, we can express the average DH intensity as

Using the expression for the phase decomposition from Eq. (1) in Eq. (6), we can derive the intensity as a function of all aberrated segment pair combinations:

This double sum combines all pairs of segments where segment $i$ has an aberration amplitude $a_i$ of the localized phase aberration $Z(\mathbf{r}-{\mathbf{r}}_i)$. These cross-terms from each aberrated pair of segments are very similar to Young interference fringes, and this forms the basic idea behind the PASTIS model.⁵⁴ The orientation and periodicity of these fringes depend on the separation and orientation of the according aberrated pair, as shown in Fig. 2.

It is important to note that the pair-wise model is not an ad-hoc idea to build the model by pairs. It derives from the fact that we expand the primary mirror phase on a discrete number of segments. Since we build a propagation model for the intensity, the “pairs” simply appear in Eq. (8) from all the cross-terms when calculating the square modulus of the electric field in Eq. (7).

Equation (8) can be readily re-written as a matrix multiplication:

While this derivation is always true in the most common case of a symmetrical DH, there are coronagraph designs that produce half-sided DHs.⁶⁷ We can show that the quadratic dependency of the contrast on the phase perturbations remains true in this most general case. We rewrite Eq. (4) in a similar matrix form as Eq. (9), but preserving the linear term:

This quadratic expression is similar to Eq. (9), but with a segmented mirror perturbation solution ${\mathbf{a}}_{\mathbf{0}}\ne 0$ that improves contrast compared to the case without aberrations. As discussed above, we also assume a wavefront control solution with deformable mirrors to be included in the term $P^{\prime }(\mathbf{r})$ (and hence $c_0$), with both amplitude and phase contributions. Therefore, this guarantees that the best contrast in the presence of that wavefront control solution and DH is obtained for ${\mathbf{a}}_{\mathbf{0}}=0$, which in turn means that any arbitrary segment aberration vector $\mathbf{a}$ will always degrade the contrast. Note that this does not preclude to have a non-zero static segmented correction included as part of the term ${\varphi }_{DH}$. This is equally true in broadband light: When summing over wavelengths, the quadratic nature of Eq. (12) remains true, albeit with different coefficients $c_0$, ${\mathbf{a}}_{\mathbf{0}}$ and $M$.

We have shown that the average DH contrast is always a quadratic function of a segmented phase perturbation ${\varphi }_s$, which can be discretized into a per-segment aberration amplitude vector $\mathbf{a}$, coefficients on a modal basis. We can calculate this average DH contrast for any aberration vector directly, using the PASTIS matrix expression [Eq. (9)]. This is particularly interesting and efficient since it does not require E2E optical simulations and only involves simple linear algebra. Furthermore, this analytical expression can be inverted to establish a segment-level WFE budget that meets a given level of contrast. This will be detailed in Sec. 4.

2.2.

Semi-Analytical Calculation of the PASTIS Matrix

The PASTIS matrix $M$ can be calculated using the original analytical approach for a perfect coronagraph, then calibrated numerically for a real coronagraph and to include pupil features (e.g., support structures) [Eq. (20) in Ref. 54]. This approach was validated against an E2E model for the 36-segment ATLAST telescope pupil with an APLC [Fig. 7 in Ref. 54] to within an error of 3%.

Here, we introduce another way to calculate the PASTIS matrix using an E2E simulation⁶⁸ of the average DH contrast for all individually aberrated segment pairs, from which we can identify semi-analytically the matrix elements in Eq. (10). This presents the advantage of enabling a direct calculation of the matrix for any telescope geometry, any coronagraph, and any choice of segment-level aberrations (including fully numerical ones such as segment figures induced by thermo-mechanical effects).

The phase for each segment pair is expressed as a Zernike aberration:

We denote by $c_{ij}={⟨I_{ij}(\mathbf{s})⟩}_{DH}$ the average DH contrast, on the pair of segments $i,j$, that can be calculated numerically for a small wavefront aberration like ${\varphi }_{ij}(\mathbf{r})$ and compared to the quadratic expression of Eq. (6) under the linear expansion of this phase term:

The elements $m_{ij}$ of the PASTIS matrix $M$ [Eq. (10)] can then be identified directly in Eq. (14) as

For simplicity, we choose the same calibration aberration amplitude $a_c=a_i=a_j$ for both segments. Throughout our analytical development above, $a_i$ is in units of radians, as $\varphi$ is a phase. Since the PASTIS matrix can be normalized to any units though, the units of the aberration amplitude $a_c$ can be chosen freely in the computation of Eqs. (16) and (17). The units of the PASTIS matrix are therefore in contrast per square of units of $a_c$ (contrast having no physical dimension), which is consistent with Eq. (9). It is noteworthy that in the presented case in Fig. 3, the units of the aberration amplitude $a_c$ is waves. The aberration amplitude $a_c$ has to be chosen such that the global pupil aberration it results in yields an average DH contrast higher than the contrast floor, but small enough to remain in the small phase aberrations linear regime. This will be discussed further in Sec. 2.3.

Fig. 3

(a) SA PASTIS matrix of the 120 segment LUVOIR-A design with the narrow-angle APLC. The matrix is symmetric by construction and the blue streaks correspond to negative values. The diagonal elements show which segments have more impact on the contrast than others. The outermost segments (60 to 120) have lower matrix values because of the darker apodization for these segments. (b) This is clearly visible in the superimposed image of the apodizer on the segments.

The matrix is symmetric by definition since $c_{ij}=c_{ji}$. Off-diagonal elements $m_{ij}$ of the PASTIS matrix [Eq. (17)] can be negative, which is not an issue since the only constraint for the matrix is to be positive semi-definite to ensure positive eigenvalues, since they correspond to each eigenmode’s contrast. This will be discussed in detail in Sec. 3.

We could potentially calculate the matrix elements $m_{ij}$ [Eq. (10)] directly by calculating those complex electric field quantities. Usually though, full E2E simulators that calculate the image plane intensity are readily available and necessary for multiple other reasons. This means that choosing to calculate the PASTIS matrix through image plane intensities makes it more flexible and portable to other simulators. More importantly, working with intensities allows us to measure an empirical PASTIS matrix without the estimation errors and computational overheads of using an electric field estimator, allowing this theory to be experimentally tested.

In summary, the PASTIS matrix is constructed in two steps: (1) calculate aberrated images $I_{ij}$ and their corresponding DH average contrast $c_{ij}$ for each pair of aberrated segments $i,j$, and (2) use these contrast values to identify analytically the elements of the PASTIS matrix $M$ based on Eqs. (16) and (17). Here, the numerical calculation of these aberrated images for pairs of segments using an E2E simulator (see Appendix A) replaces the analytical expression of Young fringes between pairs of segments.⁵⁴ This approach provides more accuracy, flexibility, and generality for use with any coronagraph and telescope geometry, since the analytical approach has to be calibrated using a numerical simulation anyway.

2.3.

Validating the Semi-Analytical PASTIS Matrix

The SA PASTIS matrix for the narrow-angle LUVOIR APLC is calculated following Sec. 2.2 and shown in Fig. 3.

The PASTIS matrix shows how some segments have a higher impact on the final contrast than others. This is visible along the diagonal, which records the contrast contribution from each individual segment alone. For example, segments 60 to 120 have a lower contrast contribution, as they correspond to the darker areas of the apodizer on the outer two rings of the aperture [see Fig. 3(b)]. This effect is also visible on the innermost ring of hexagons. We can also notice streaks of negative values in the matrix in the off-axis areas, as discussed in Sec. 2.2.

We validate the SA PASTIS matrix by comparing the PASTIS contrast obtained with the matrix formalism of Eq. (9) to the contrast from an E2E simulator using the same inputs. We show the comparison in Fig. 4. The coronagraph floor for this particular APLC design in the absence of aberrations is $4.3\times {10}^{-11}$. The PASTIS model starts to diverge from the E2E calculation at large WFE root-mean-square (RMS) where the linear approximation of the phase breaks down. It is noteworthy that the choice of $a_c=1/500$ of the wavelength (used in the presented example, and is on the order of 1 nm in the visible range) on a single segment yields a global pupil WFE RMS of $1.67\times {10}^{-5}$ waves, which translates into an average DH contrast just above the coronagraph floor, but keeps it in the small aberration regime. Here, the accuracy of the SA matrix approach is significantly higher than that of the fully analytical matrix because the construction of the PASTIS matrix is based on the actual E2E simulation as opposed to a post-calibrated analytical fringe model.

Fig. 4

Average DH contrast as a function of WFE using both an E2E simulator (dashed red) and the PASTIS matrix propagation (solid blue). In a hockey stick graph behavior, the contrast is limited by the coronagraph itself at low WFEs corresponding to the flattened-out curve to the left (at $c_0$). From about ${10}^{-4}$ waves to ${10}^{-1}$ waves of WFE RMS, the contrast is limited by segment phasing aberrations. In this range the estimation error of PASTIS is 0.06% compared to the reference E2E model. A calibration aberration per segment $a_c$ of 1/500 wave, on a 120 segment pupil, translates to a global WFE of $1.67\times {10}^{-5}$ waves when calibrating the PASTIS matrix diagonal, or $2.36\times {10}^{-5}$ with two simultaneously aberrated segments, which is in the small aberration regime of the model, just above the coronagraph floor. The curves shown are obtained as the mean of the same 20 random realizations for each RMS value, both for the E2E simulator and the PASTIS propagation. At large WFEs (close to 0.1 waves RMS) the linear approximation breaks down and the two curves no longer match perfectly.

3.1.

Eigendecomposition of the PASTIS Model and Mode-Segment Relationship

The PASTIS matrix $M$ is square and symmetric by construction, and therefore diagonalizable. We perform the eigendecomposition:

Fig. 5

All PASTIS modes for the LUVOIR-A narrow-angle APLC, for local piston aberrations, sorted from highest to lowest eigenvalue. The modes are unitless, showcasing the relative scaling of the segments to each other, and between all modes. They gain physical meaning when multiplied by a mode aberration amplitude $b_p$ in units of WFE or phase. Their respective eigenvalues and hence relative impact on final contrast is shown in Fig. 6.

Fig. 6

Eigenvalues, or sensitivity of contrast to mode index $p$, for the piston PASTIS modes of the LUVOIR-A telescope with the small FPM coronagraph design, shown in Fig. 5. Note how the PASTIS matrix and modes do not depend on the target contrast, but they do on the choice of telescope geometry and coronagraph, making them the proper modes of the optical system.

Fig. 7

Low-impact modes with high tolerances for the narrow-angle APLC on the LUVOIR-A telescope, for local piston aberrations. These modes have little impact on the final contrast—they are similar, but not equal, to discretized Zernike modes and the coronagraph rejects them very well by design.

Fig. 8

Mid-impact modes with medium tolerances for the narrow-angle APLC on the LUVOIR-A telescope, for local piston aberrations. These modes have medium impact on the final contrast, relatively speaking. These modes show mostly low spatial frequency features except for high spatial frequency components in the parts of the pupil where the apodizer covers most of the segments.

Fig. 9

High-impact modes with low tolerances for the narrow-angle APLC on the LUVOIR-A telescope, for local piston aberrations. These modes have the highest impact on the final contrast. They consist entirely of high spatial frequency components in the parts of the pupil where the apodizer (and other pupil plane optics) are the most transmissive.

The eigenvalues ${\lambda }_p$ shown in Fig. 6 indicate how much each mode contributes to the final image contrast if applied to the pupil in their natural normalization, without any imposed weighting. This figure shows that the high-spatial frequency modes (to the left) have a much higher impact than the lower-spatial frequency modes (to the right).

The PASTIS modes ${\mathbf{u}}_p$ form an orthonormal basis set that allows us to express any arbitrary, segment-based pupil plane aberration $\mathbf{a}$ as a linear combination of the modes ${\mathbf{u}}_p$ with mode weighting factors $b_p$:

This can also be written as

3.2.

Contrast as a Function of the Eigenmodes

The mode weights $\mathbf{b}$ will depend on how much each individual mode contributes to the final contrast, and their associated eigenvalue. Inserting Eq. (20) into Eq. (9) allows us to define this relationship:

The final contrast is therefore the sum of all squared mode weights, multiplied by their respective eigenvalue. Since the modes contribute independently to the final contrast (they are orthonormal by construction), we can define a per-mode contrast as

We can then find the $p$’th mode weight that gives the allocated contrast contribution $c_p$ as

Equation (25) gives the weighting factor for each PASTIS mode when it has a particular contrast contribution $c_p$. We can illustrate this expression by calculating the mode weights corresponding specifically to a uniform contrast contribution of the overall target contrast $c_t$ over all modes, $c_p=(c_t-c_0)/n_{\text{modes}}$. Then we calculate them as

Fig. 10

PASTIS mode weights for the uniform contrast allocation across all modes. The low-index modes to the left, which correspond to high spatial frequencies, have a lower WFE tolerance than the low spatial frequency modes with high index to the right. These mode amplitudes are inversely proportional to the eigenvalues associated with each mode [Eq. (26)], and they scale the modes such that each of them contributes the same contrast $c_p$ to the overall target contrast. The cumulative contrast response of the modes multiplied by these weights is shown in Fig. 11.

In Fig. 11, we confirm the validity of the mode weights $\widetilde{b_p}$ by showing the average DH contrast from an E2E propagation of the cumulative WFE for all modes. The linearity of the plot, as well as the end value at the target contrast $c_t$ validates the uniform contrast allocation to each PASTIS mode from Eq. (26).

Fig. 11

Cumulative contrast from all PASTIS modes when allocating the total contrast uniformly across all modes (see Sec. 3.2). For instance, the measured contrast corresponding to the first 60 accumulated weighted modes is about $0.7\times {10}^{-10}$. We multiply all modes by their respective mode amplitude $\widetilde{b_p}$ and propagate them cumulatively to the image plane, both with the E2E simulator (dashed red) and the PASTIS propagation (solid blue). Without any of the modes applied, we get the contrast floor from the coronagraph $c_0$, while application of all modes together yields the requested target contrast, here $c_t={10}^{-10}$. Each mode is allocated an equal contrast contribution $c_p$ to the final contrast, which results in a linear cumulative contrast curve. Note how neither line starts at the coronagraph floor because the lowest-index mode already adds a contrast contribution on top of the baseline contrast. The corresponding PASTIS mode weights to obtain this uniform allocation of contrast per mode, is shown in Fig. 10.

3.3.

Statistical Mean of the Contrast from Mode Amplitudes

In this section, we analyze the properties of the model in the statistical sense to prepare a framework for the segment-level error budget in Sec. 4. We extend the formalism from purely deterministic mode weights $\mathbf{b}$ to random variables. We obtain the statistical mean contrast by substituting Eq. (23) into Eq. (24) and taking the mean:

Assuming zero-mean normal distributions of the PASTIS modes, we can readily identify their variance as

We verify this for the uniform contrast allocation per mode [Eq. (26)] in an E2E MC simulation where we draw random samples of the PASTIS mode coefficients $\tilde{\mathbf{b}}$, following zero-mean normal distributions $\mathcal{N}$ with standard deviations ${\sigma }_p$, according to the uniform contrast allocation from Eq. (26):

Figure 12 shows the result of such an MC simulation where we sum each randomly weighted, individual set of modes to a unique wavefront map, propagate it to the image plane with the E2E simulator and measure the spatial average contrast in the DH. We validate that the statistical mean of all these contrast values is the target contrast $c_t$ for which we calculated the vector of mode standard deviations $\mathit{\sigma }$ in the first place, in this case ${10}^{-10}$.

Fig. 12

Validation of the uniform contrast allocation across PASTIS modes with an E2E MC simulation, drawing random sets of mode weights from a normal distribution with standard deviations $\mathit{\sigma }$ [Eq. (28)], corresponding to an equal contrast allocation per mode [Eq. (26)] with weights $\tilde{\mathbf{b}}$. The overall WFE of one realization is the sum of all weighted modes for each set, and is propagated in an E2E simulation. The histogram represents 100,000 realizations of the average DH contrast for a target contrast of ${10}^{-10}$.

In summary, PASTIS provides an analytical model to go from a set of segment aberrations to the DH average contrast in the coronagraphic image. The calculation of the PASTIS matrix eigenmodes allows to invert this model: we can set a target contrast and allocate WFE amplitudes to each of the system’s eigenmodes to reach that target contrast. Since they form an orthonormal set of modes, their individual contrast contributions add independently, each with its own sensitivity, as a fraction of the average contrast in Sec. 3.2, on top of their statistical description in Sec. 3.3. These optical modes contain the full information of the image formation system, including the apodizer, coronagraph, primary geometry, and other optical components in the system, and are distinct from and unaware of the mechanical behavior of the telescope. By this nature, we can use them to understand the fundamental limitations for high contrast imaging with a segmented aperture, which will in sequence be further constrained by thermo-mechanical properties of the telescope as we describe in Sec. 4.

4.1.

Statistical Mean Contrast and Its Variance in Segmented Coronagraphy

We can calculate the statistical mean contrast of the DH spatial average directly from Eq. (9), exploiting the fact that the trace of a scalar is the scalar itself, and that $\mathrm{tr}(AB)=\mathrm{tr}(BA)$:

Similarly, we can derive an analytical expression for the variance $\mathrm{Var}(c)$ of the DH contrast. Assuming that $\mathbf{a}$ follows a zero-mean Gaussian distribution, the variance for Eq. (9) takes the very simple form⁶⁹ (Theorem 5.2c):

These two equations provide an unambiguous closed form derivation of the mean contrast and its variance from the optical model of the imaging system (encapsulated in $M$), and from the thermo-mechanical properties of the telescope (captured by $C_a$). The PASTIS matrix $M$ knows nothing of the thermo-mechanical effects of the observatory and is obtained by diffractive modelling of the coronagraph, while the segment covariance matrix comes from thermal and mechanical modeling of the observatory and is completely detached from the image formation system of the telescope. The two matrices together ($M$ and $C_a$) fully describe the statistical response of the coronagraph system to a particular WFE allocation on segments and therefore allow to establish a set of top-level requirements on segment tolerances for an observatory.

The enabling aspect of Eqs. (31) and (32) for segment-level tolerancing is that the trace is invariant under a basis transformation. It follows that if either one of the two matrices, the PASTIS matrix or the thermo-mechanical covariance matrix, is expressed in its diagonal basis, the expressions for the contrast mean and variance simplify greatly, as we show in the following sections. The segment tolerancing can thus be achieved either by diagonalizing $M$, or by doing so with $C_a$. We have treated the case of diagonalizing the PASTIS matrix $M$ in Sec. 3, where we describe the analytical framework for segmented telescope tolerancing in the diagonal basis that most naturally describes the optical sensitivity of the system to the DH contrast. In the following two sections, we turn to a basis that diagonalizes the segment covariance matrix instead, permitting us to perform the tolerancing on appropriate system modes.

4.2.

Uncorrelated Segment-Level Requirements

In finding a diagonal basis for the thermo-mechanical matrix, the easiest case is when $C_a$ is already diagonal, which physically corresponds to independent segments on the primary mirror. In this case, the diagonal elements of $C_a$, namely, the segment variances $⟨a_k^2⟩$, fully describe the effect of the primary mirror segments on the DH contrast, and the statistical mean of the contrast in Eq. (31), $⟨c⟩$, finds a simple expression similar to Eq. (27):

Similarly to the mode-based error budget presented in Sec. 3, we now want to find a segment-based error budget to formulate the WFE limits on each segment that reach a specific statistical mean target contrast $c_t=⟨c⟩$. Turning to a statistical mean contrast allows us to define a similar allocation of contrast contributions to all segments as we did statistically (and deterministically) in the PASTIS mode basis [Eq. (24)]. The most straightforward way of doing this is to allocate the target contrast equally to all segments:

If we define ${\mu }_k$ as the standard deviation of the WFE on the $k$’th segment, comparably to Eq. (28):

The expression in Eq. (36) lets us calculate a per-segment requirement for all individual segments in the pupil of a coronagraph instrument, given a statistical mean target contrast. The main assumption for this is that the segments are independent from each other, and that we have access to the statistical mean value of the contrast. While the mean contrast is easily measurable on images through averaging, this might not be the case for an ultra-stable facility such as LUVOIR. However, the statistical mean contrast is an important quantity to perform segment-level WFE tolerancing, especially with regards to mirror manufacturing.

In a more physical sense, we know that the intensity or contrast is proportional to the variance of the WFE. Therefore, the total final contrast over the full pupil is proportional to the sum of the segment variances, which is also proportional to the sum of the contrast for all segments, conforming with Parseval’s theorem. We validate this independent segment-level error budget for three different LUVOIR coronagraphs in Sec. 5.

4.3.

Case of Correlated Segments

While the assumption of statistically independent segments brings insights into WFE tolerances of segmented mirrors for coronagraphic imaging, it is not general enough to encompass all possible modes for such telescopes where large-scale thermo-mechanical drifts occur (for example, backplane “flapping” mode around the folding motion of the primary mirror). Here, we discuss extensions of the PASTIS approach to the case of correlated segments.

In the case of independent segments, the covariance matrix $C_a$ of the aberration vector $\mathbf{a}$ is a simple diagonal matrix holding the segment variances ${\mu }_k^2$. However, the covariance matrix $C_a$ is no longer diagonal for correlated segments, because of mechanical coupling for example due to large-scale backplane deformations. In this case, the statistical mean contrast and its variance remain analytically computable with Eqs. (31) and (32). The tolerancing can be done by performing an eigendecomposition on $C_a$, which will diagonalize it and provide an orthonormal set of eigenmodes that describe the mechanical perturbations of the telescope system, which is also known as the Karhunen–Loève basis. By writing $C_a=V{C}_{th}{V}^T$, we obtain the PASTIS matrix in this new basis, $M^{\prime }$, through the transformation matrix $V$, where $M^{\prime }=V^{T}MV$ ($m_{kk}^{\prime }\in M^{\prime }$) still describes the optical properties of the system. In this basis, Eq. (31) takes the form:

Since the thermo-mechanical covariance matrix $C_{th}$ is diagonal, we identify its diagonal elements as the thermo-mechanical mode variances $s_k^2$. Like for Eqs. (27) and (33), this simplifies Eq. (37) yet again to

In this most general case of correlated segments, the knowledge of the segment-level covariance matrix $C_a$ and its diagonal eigenbasis $C_{th}$ supersedes the simpler description in terms of segment-level variances that is only possible in the uncorrelated case (Sec. 4.2). It allows us to express the mean contrast explicitly as a function of variances of thermo-mechanical modes that can be toleranced in a similar fashion to what was done in Sec. 3.

We have presented a quantitative, fully analytical method to calculate segment-level tolerances for a high-contrast instrument on a segmented aperture telescope. These follow directly from the PASTIS matrix for which we provided a new, SA way for its calculation that exploits a numerical simulator to compute the effects of the segments on the intensity in the image plane. We encode the optical and thermo-mechanical properties of the observatory separately, with the PASTIS matrix $M$ and the segment covariance matrix $C_a$, which when put together allow for the analytical calculation of the expected mean contrast and its variance. These equations are invariant under a basis transformation, which permits us to find an appropriate diagonalized basis to derive individual WFE tolerances. This can either be done by diagonalizing $M$, as we showed in Sec. 3, or by finding a diagonal basis for $C_a$. A special case is given if $C_a$ is naturally diagonal due to independent segments on the segmented mirror; in this case, we derive a per-segment requirement map by following the analytical framework set forth in Sec. 3. In the more general case of correlations between the segments, due to thermo-mechanical properties of the telescope, we diagonalize $C_a$ and use the same analysis principles in the Karhunen–Loève basis of $C_a$, which allows us to calculate per-mode WFE tolerances. While the deformation matrix $C_a$ will be acquired through thermo-mechanical modeling and can include thermal, vibrational, or gravitational perturbations, the PASTIS matrix $M$ and the PASTIS modes give insight into the purely optical properties of the observatory, and the sensitivity of the optical system to contrast.

In the next section, we validate the segment-level error budget in the case of independent segments (Sec. 4.2) for three different APLC designs for LUVOIR.

5.1.

Segment Requirements and Monte Carlo Simulations for Three APLC Designs

We first calculate the PASTIS matrix for each of these three APLC designs, according to the methodology described in Sec. 2.2. We can then establish a segment-level error budget in the assumption of uncorrelated segments, according to Eq. (36). In Fig. 13, we show the resulting segment requirement maps for a target contrast of $c_t={10}^{-10}$ and for all three APLC designs.

It is important to note that these requirement maps do not represent the WFE over the segmented pupil, but instead show the standard deviations on the tolerable WFE for each segment to retrieve, statistically, the desired mean target contrast. In this sense, the maps in Fig. 13 are a prescription for the drawing of random segment WFE realizations such as the examples shown in Fig. 14. These random maps are then propagated with the E2E simulator and their average contrast values build the MC histograms in Fig. 15. One big takeaway point from Fig. 13 is that the segment requirements are not uniform across the pupil, but clearly follow the apodization of the coronagraph mask. The PASTIS matrix holds knowledge of the optical effect of not only the segments but also the coronagraph instrument on the final contrast, so by including that knowledge into the derivation of the segment constraints we obtain a requirement map optimized for that particular instrument. Moreover, we can observe a direct trade-off between the coronagraph apodization and the per-segment requirements—the more aggressive the apodization and the lower the throughput, the more we can relax the requirements on the more concealed segments within one coronagraph. However, more aggressive apodization usually comes with smaller FPM coronagraphs that filter low-order modes less, which will lead to more stringent overall requirements. This leads to a direct trade-off between FPM size, throughput and WFE requirements (see also Sec. 6).

Fig. 14

Four random segment-based WFE maps drawn from a zero-mean normal distribution and the per-segment standard deviations from the left prescription map in Fig. 13, for the narrow-angle APLC design and a target contrast of ${10}^{-10}$. After each random map is created, we propagate it through the E2E simulator and record its average contrast to build the left MC simulation in Fig. 15.

Fig. 15

Validation of the independent segment tolerancing with E2E MC simulations, for different target contrasts, using the narrow-angle APLC design. Each segment $k$ in one of the 100,000 WFE realizations is drawn from a zero-mean normal distribution with standard deviation ${\mu }_k$. The dashed-dotted lines mark the target contrast of each case, which are successfully recovered by the mean values of the histograms, in accordance with their analytical calculation in Eq. (31). The dotted lines mark the 1-sigma confidence limits of this contrast distribution, which are $8.3\times {10}^{-12}$, $1.4\times {10}^{-10}$ and $1.5\times {10}^{-9}$ for the three target contrasts ${10}^{-10}$, ${10}^{-9}$ and ${10}^{-8}$ respectively, and they accord with the numbers calculated by Eq. (32).

These requirement maps can be calculated for any target contrast in the range of validity of the PASTIS model, which we discussed in Sec. 2.3. We can verify Eq. (36) by running MC simulations with the E2E simulator across a grid of different coronagraph instruments and target contrasts. Using a range of target contrasts $c_t={10}^{-10}$, ${10}^{-9}$, and ${10}^{-8}$ on the narrow-angle baseline LUVOIR APLC design, we first calculate the segment constraints (the requirement maps for ${10}^{-9}$ and ${10}^{-8}$ are not shown here, but they show the same spatial distribution over the segments as in Fig. 13, only different by a proportionality factor). We draw the WFE amplitude for each individual segment from a zero-mean normal distribution and its standard deviation ${\mu }_k$:

We use these random aberration amplitudes on all segments to compose a WFE map on the segmented pupil and then propagate this WFE map through the E2E simulator to measure the resulting spatial average contrast in the DH. Doing this 100,000 times for each target contrast case, we obtain the histograms shown in Fig. 15. The mean of the resulting MC simulations clearly recovers the target contrast for which the segment requirements have been calculated, which is indicated by the dashed-dotted line. Both these mean values, and the standard deviations, indicated with the dotted lines in Fig. 15, agree with the theoretical values calculated analytically from the segment covariance matrix [Eqs. (31) and (32)]. Also, we have verified the correct recovery of the same range of target contrasts by means of MC simulations for the other two APLC designs shown in Fig. 13 (resulting histograms not shown in this paper).

5.2.

Modal Analysis of the Segment-Based Requirements

The segment requirement maps were obtained assuming a uniform contrast allocation across all segments [Eq. (34)]. We also assumed statistically independent segments, so that their correlation matrix $C_a$ was diagonal. Here, we further explore this uniform error budget in the segment basis by analyzing the corresponding distribution of proper system modes of the optical system, the PASTIS modes. Using the transformation matrix $U$ from the eigendecomposition of the PASTIS matrix $M$, we can calculate the corresponding covariance matrix in the PASTIS mode basis with $C_b=U^T{C}_{a}U$. Given this linear transformation, if the covariance matrix is diagonal in one space, we do not expect it to be diagonal in the other space. The matrix $C_b$, obtained from the diagonal segment covariance matrix $C_a$ assembled from the requirement map, is shown in Fig. 16(a). Figure 16(b) compares the extracted standard deviations for PASTIS modes along the diagonal of $C_b$, with the PASTIS mode weights previously calculated in Sec. 3. Although $C_b$ is not diagonal, this is a legitimate comparison. The PASTIS matrix $M$ is always diagonal in its own eigenbasis, expressed as matrix $D$ in Sec. 3.1. This is why the average contrast expression from the statistical mode weights in Eq. (27) only requires the diagonal elements of the covariance matrix $C_b$, no matter whether it is diagonal or not, i.e., whether the mode weights show some correlations or not. The difference with respect to the mode weights $\widetilde{b_p}$ [Eq. (26)] obtained under the assumption of a uniform contrast allocation per mode (Fig. 10) is notable: the mode weights of low mode index have increased tolerances, which is very interesting from a system design point of view, while large index modes (above $\sim 90$) are strongly attenuated in the PASTIS mode basis error budget for independent segments. This is also clearly visible in Fig. 17 where the contrast contribution per mode is relatively flat at a low mode index but drops to negligible contributions at high-index modes. For the case of the flat contrast allocation across modes, this figure shows a flat line at $(c_t-c_0)/n_{seg}$ for comparison (dashed line). This effect can be well understood by looking back at Fig. 5, where high-index modes appear to be very similar to low-order Zernike modes, therefore, having highly correlated segments, and the low-index modes appear as high-spatial frequencies, i.e., with more uncorrelated segments. Therefore, the construction of a segment-level error budget for uncorrelated segments creates a modal distribution with extremely low weights on the PASTIS modes that have highly correlated segments (high-index modes), as seen in Fig. 16. Also, since the mode contrast contribution is directly related to the mode weight [Eq. (25)], the same effect is visible in the allocated contrast per mode (Fig. 17).

Fig. 16

(a) Covariance matrix $C_b$, calculated from the diagonal covariance matrix in segment-space, $C_a$, with $C_b=U^T{C}_{a}U$. Although there are clearly some correlations present between the high-index PASTIS modes in the top right corner (low spatial frequencies), this does not matter as long as we are in the PASTIS segment basis, where the PASTIS matrix is diagonal. When this is the case, the mean contrast only depends on the diagonal elements of $C_b$ [Eq. (27)]. (b) PASTIS mode amplitudes for the case of independent segments in WFE RMS (solid red). They are extracted from the mode covariance matrix $C_b$, after constructing an error budget assuming independent segments that contribute equally to the total contrast, at 500 nm. Overlapping (dashed gray), we can see the mode weights from the uniform contrast allocation to all PASTIS modes from Fig. 10. We can clearly see how compared to that flat allocation, the independent-segment error budget increases the tolerances of low-index modes (left) that have less segment correlation, and dampens the tolerances of high-index modes (right) that are highly correlated, low-spatial frequency modes.

Fig. 17

Contrast per individual PASTIS mode when derived from the error budget in which all segments contribute independently and equally to the final contrast (solid blue). High-index modes to the right, which correspond to low-spatial frequencies and therefore highly correlated segments (see Fig. 5), are highly attenuated and contribute negligible amounts to the contrast. The uniform contrast allocation across all modes at $(c_t-c_0)/n_{seg}$ is indicated with the dashed gray line (the contrast floor has been removed in both curves).

To illustrate this further, we calculate a cumulative contrast plot similar to Fig. 11, which was initially obtained for a uniform contrast allocation per mode. The new result is shown in Fig. 18, where we can see that it is no longer linear, i.e., the modes no longer contribute equally to the total mean contrast. The slope of the blue curve is indicative of the allocated tolerances for each mode: the low-spatial frequency PASTIS modes on the righthand side now contribute significantly less to the final contrast, while the first $\sim 80$ modes contribute more, while still resulting in the exact cumulative target contrast. This is consistent with the behavior discussed in Figs. 16 and 17.

Fig. 18

Cumulative contrast of the PASTIS modes for the error budget in which all segments contribute independently and equally to the final contrast (solid blue), compared to the case in Fig. 11 where all modes contribute the same contrast (dashed gray). The high-index modes have negligible contrast impact (see also Fig. 17) as they correspond to low-spatial frequency, highly correlated segments. This plot also confirms the assumption that the mode covariance matrix $C_b$ is nearly diagonal.