3.1.

Core Throughput

A key attribute of a coronagraph is its core throughput. The planet PSF’s “core” is defined as the area circumscribed by its half-max contour. Core throughput is the fraction of the planet light collected by the telescope primary mirror that ends up inside the core region (see Fig. 8). The photometric SNR is influenced most strongly by the high-signal part of the PSF, and the core is a good representation of that domain. The solid angle subtended by the core ${\mathrm{\Omega }}_{\text{core}}$ depends on the aperture. For an unobscured circular collector (primary mirror) and no coronagraph, the PSF core is a circle of diameter $\sim 1\lambda /D$.

Fig. 8

The PSF for a vector-vortex coronagraph charge-6 (VVC-X6), when the source is 3 $\lambda /D$ from the nominal LOS. The core area is enclosed by the half-max (0.5) contour. This core area corresponds to some solid angle ${\mathrm{\Omega }}_{\text{core}}$ on the sky. The PSF is asymmetric because of the proximity of the source location to the IWA of the coronagraph, which is 2.3 $\lambda /D$ in this case.

Core throughput includes two effects: the loss of light due to partial or complete obscuration by the masks (particularly the FPM) and the spread of the PSF beyond the core boundary. Both of these are diffractive effects. In searching for Earth-like planets, to get a sufficient sample of Sun-like stars, one must search as large a volume of space as possible. This, in turn, drives the need for good performance at smaller working angles. The IWA is one metric that is sometimes used for this purpose.

3.2.

Inner and Outer Working Angles

The IWA is defined as the angular separation from the line of sight (LOS) below which the azimuthally averaged core throughput falls below 1/2 of its maximum value within the dark hole. As an example, Fig. 9 shows the azimuthally averaged core throughput for the vector-vortex coronagraph, charge-4 case. The maximum throughput is seen to be 38%. The IWA for this coronagraph is a very favorable $1.6\lambda /D$, but it has good throughput ($>5\%$) all the way down to $1\lambda /D$.

Fig. 9

Example of azimuthally averaged core throughput. Shown is the vector-vortex coronagraph, charge-4 case. The maximum throughput is about 38%. This is indicated by the orange, dashed horizontal line. Another such line, in green, indicates the half-max level, and the IWA is indicated by the intersection point of the throughput curve and the half-max line. The IWA is seen to be $1.6\lambda /D$.

The existence of an OWA can be inherent to the coronagraph architecture (for example, if the FPM is designed with an annular opening), but, even when it is not, it is practically limited by the number of actuators in the DM. This is because the DM must always be used for high contrast, at least to compensate for the optics imperfections.

3.3.

Raw Contrast

The coronagraph attribute to compare with the planet flux ratio is its raw or initial contrast. The qualifier raw is used to distinguish it from the residual contrast after differential imaging and other postprocessing. Unless otherwise indicated, we will henceforth use the unqualified form to refer to raw contrast.

Contrast is the measure of the effectiveness of the coronagraph in suppressing starlight near the planet. Considering an angular location $(u,v)$ within the field of view, the raw contrast is the ratio of the starlight scatter throughput to that point, over the planet throughput at that point.

For a star located at the nominal LOS of the instrument, angular coordinates (0, 0), the throughput to $(u,v)$ is given by the fraction of the incident light from the light source that ends up within some region of interest ${\mathrm{\Omega }}_r$, centered at $(u,v)$. By “the incident light,” we refer to the total power incident on the usable, unobscured portion of the collecting aperture, which is the telescope primary mirror. We label this throughput as $\tau (u,v)$: a quantity evaluated at $(u,v)$, with the source at (0, 0). The reference region of interest ${\mathrm{\Omega }}_r$ can be thought of equivalently as a two-dimensional area on the image plane or a solid angle on the sky. In hardware, the width of ${\mathrm{\Omega }}_r$ is typically chosen to be that of a detector pixel. In modeling, it is chosen to be that of a modeling pixel. The exact choice is not critical as long as it is small compared to $\lambda /D$. For a planet located at $(u,v)$, the throughput is simply the fraction of the incident light that is detected within ${\mathrm{\Omega }}_r$. We label this as ${\tau }_{pk}(u,v)$: i.e., the throughput into a reference region centered at $(u,v)$, with the source located also at $(u,v)$. Contrast at $(u,v)$ is simply the ratio of the two throughputs:

Evaluation of contrast is inherently cumbersome: to map out the contrast one needs to evaluate the denominator, which means placing a test source (whether in a hardware test or in computer modeling) at a grid of $(u,v)$ points within the relevant part of the field of view. In a testbed, this means the incoming beam is tilted with a mirror so that it now appears to come from $(u,v)$, while in modeling the incoming wavefront phase is given a tilt of $(u,v)$. By comparison, the numerator is obtained from a single image, whether in hardware or in modeling. Because of the time-consuming nature of measuring contrast, a simplified approximation is often computed, called normalized intensity (NI). Its definition is very similar to contrast:

Henceforth, the term working angle (designated by $\alpha$) will be used to refer to the separation angle between a point of interest (such as a planet location) and the LOS in $\lambda /D$ units: $\alpha =\sqrt{u^2+v^2}/(\lambda /D)$. Also a subscript of $\alpha$(such as in $C_{\alpha }$ defined above) shall imply an azimuthally averaged quantity where the remaining dependence is on the radial distance $\alpha$ in the dark hole. For a radial band of width $\delta \alpha$ centered on working angle $\alpha$, the azimuthally averaged contrast is given by

Fig. 10

The importance of using contrast over NI is shown here by plotting the azimuthally averaged synthetic contrast $C_{\alpha }$ over ${\mathrm{NI}}_{\alpha }$ for the different coronagraph cases as a function of working angle. The legend indicates the IWA for each case. Near the IWA, the ratio $C_{\alpha }/{\mathrm{NI}}_{\alpha }$ is seen to be quite significant.

The appeal of using (NI) instead of contrast $C$ is that it replaces hundreds of propagations (each with a different incoming wavefront) with a single operation (in hardware) or propagation (in software). For example, as shown in Fig. 11, to evaluate the contrast from 2 to $9\lambda /D$ in radial steps of $0.3\lambda /D$, a total of 1703 separate “pointings” ($u_i,v_i$) are needed, if 2 azimuthal samples per $\lambda /D$ are used. The disadvantage of NI is that it tends to underestimate the contrast at small working angles. This is because contrast accounts for throughput loss for imaging off-axis sources—i.e., planet throughput (see core throughput below)—while NI does not (due to the FPM being removed). The difference between $C$ and NI diminishes as the source working angle $\alpha$ becomes larger than the IWA by a few $\lambda /D$. Conveniently, it is just as NI becomes a good approximation that calculating $C$ becomes most cumbersome. The number of pointings $(u,v)$ is small at the most important, smaller working angles $\alpha$, while it grows as $2\pi \alpha$ as $\alpha$ is increased. This fortuitous condition can be exploited by defining a “synthetic contrast, $C_{\mathrm{syn}}$,” which equals contrast near the IWA, equals NI near the OWA, and transitions at some intermediate working angle ${\alpha }_t$ between the two, given by

Fig. 11

Example of a grid of pointing offsets $(u,v)$ for evaluating contrast and core throughput.

Figure 12 shows contrast, NI and synthetic contrast for the VVC-6 case. The IWA for this coronagraph is $2.3\lambda /D$, and, as can be seen from the plot in Fig. 10, NI underestimates $C$ by a factor of $\sim 4$ near the IWA. The discrepancy differs for other coronagraph cases, but this case illustrates how important the distinction is between $C$ and NI. In the literature, sometimes the distinction between $C$ and $NI$ is not made, and the quantity called “contrast” is, in fact, NI. But leakage should be measured in $C$, and $NI$ does not approximate $C$ well enough near the all-important IWA. The IWA is usually the region of greatest interest for exoplanet searches.

Fig. 12

Azimuthally averaged values of contrast (solid), NI (dotted), and synthetic contrast (dashed) for the vector-vortex coronagraph, charge-6 case. The transition working angle is chosen near $9.5\lambda /D$.

For all cases of interest $C\ll 1$ and as a result, it can be shown that the contrast $C$ is proportional to the square of the wavefront error $w$ (i.e., $C\propto w^2$). The proportionality factor depends on the wavefront error mode and the coronagraph design (masks and DM configuration).

3.4.

Comparing Core Throughput and Raw Contrast among Architectures

So far, we have shown plots of core throughput and contrast in the more common $\lambda /D$ units. However, comparisons in these units are inconvenient. For example, at 550 nm, $\lambda /D$ is $\sim 28\mathrm{mas}$, for $D=4\mathrm{m}$ and $\sim 19\mathrm{mas}$ for $D=6\mathrm{m}$. Hence, the exo-Earth at 100 mas separation would be located at $\sim 3.6\lambda /D$ for a 4-m telescope and $\sim 5.3\lambda /D$ for a 6-m telescope.^27,28 To facilitate meaningful comparison of candidate architectures, it is better to plot core throughput in scientifically relevant angular separation units. Figure 13 plots core throughput versus angular separation for four cases: three different coronagraphs (VVC-4, VVC-6, and HLC) on a 4-m off-axis monolithic aperture telescope and an APL coronagraph on a 6-m on-axis segmented aperture telescope. Note that Fig. 13 excludes throughput losses other than core throughput. Losses, such as those from reflection off mirrors or transmission through filters, are bookkept in the photometric calculations. Two key takeaways from Fig. 13 are the following: (1) different coronagraphs on the same telescope have significantly different core throughputs and (2) the central obscuration of an on-axis telescope greatly reduces core throughput.

Fig. 13

Comparison of core throughput versus separation angle for five cases (three coronagraphs with a 4-m off-axis monolithic and two 6-m segmented cases). The 6-m off-axis case with a VVC6 has the same core throughput profile as the 4-m off-axis monolith with VVC4. The separation for an exo-Earth at 10 pc (100 mas) is indicated with the vertical line. Note that while the 6-m (on-axis) segmented aperture has $\sim 2\times$ more collecting area than the 4-m aperture, its “comparative” throughput at 100 mas is only $\sim 5\%$.

It is similarly helpful to plot raw contrast versus angular separation. But the value of the raw contrast achieved depends on wavefront control performance. In keeping with the modeling approach outlined in Sec. 4, we use a per-design specified wavefront plus an additional postwavefront control surface error. There is sufficient difference between the raw contrast of the different architectures to reliably distinguish the two vector-vortex cases as a group from the HLC case and the APLC case. Figure 14 shows the raw contrast versus angular separation for each case with an assumed postcontrol residual surface error of 120 pm rms (picometers root-mean-square). Again, the key takeaways are that different coronagraphs perform differently and that a central obscuration significantly degrades performance. Also the off-axis unobscured monolithic aperture with a vector-vortex coronagraph has the smallest IWA.

Fig. 14

Comparison of raw contrast versus separation angle for four architectures. Raw contrast depends on wavefront control, but here we instead assumed a postcontrol residual surface error of 120 pm rms. Notice that the vortex coronagraphs, particularly charge-4, have much smaller IWAs and a reach that is many times that of the obscured segmented case.

4.1.

Flux Ratio Noise as the Error Budget Metric

As described in Appendix A, the planet (electron) count rate is given by

Although the signal $S$ consists of photoelectrons at the detector over some integration time $t$, the noise comes from a variety of sources. We enumerate these as: (1) shot noise in the planet signal (${\sigma }_{\mathrm{pl}}$), (2) shot noise in the underlying speckle (${\sigma }_{\mathrm{sp}}$), (3) shot noise in the underlying zodiacal dust background (local + exo) (${\sigma }_{\mathrm{zo}}$), (4) detector noise (${\sigma }_{\det }$), and finally (5) the residual speckle instability error ${\sigma }_{\mathrm{\Delta }I}$. The total variance is given by

If ${\sigma }_{\mathrm{tot}}$ is the total noise associated with the signal $S$, then the noise in measuring the planet flux ratio is given by

4.2.

Strawman Observing Scenario for the Error Budget

Having identified the science objective and the error metric, the next step is to form a representative observing scenario. This is also called a “strawman” scenario in the sense that it may not correspond to any actual observation in perfect detail, but contains enough of the aspects of the expected observations to fill the convenient role of a single operating concept for quantitative analysis.

For our target case, the scenario includes a certain period of staring, pointed at the host or target star. The coronagraph operates to suppress unwanted starlight and any system errors. Some filtering is employed in ultrahigh-precision measurements to assist in driving down the errors. For example, if the errors were strictly random (such as photon noise or detector noise), simply extending the observation duration $t$ would reduce the relative error at a rate of $1/\sqrt{t}$. This by itself would call for a long integration time in the observing scenario. However, in a real instrument, unsensed drift errors, possibly from thermal sources, begin to dominate as integration times are lengthened. The effect of these drift errors is variation in the starlight residual speckle (i.e., contrast instability). Some new technical innovations are currently being developed for sensing speckle instability in real time,²⁹ but as of this writing it remains to be seen whether they will be feasible. At present, the best understood method of mitigating speckle instability is via chopping, where a measurement is taken of a reference star and subtracted from the target star measurement. In the context of coronagraphs, this is called differential imaging. Another form of differential imaging, called angular differential imaging, is based on observing the same star at different roll angles. For either method, if the speckle subtraction is perfect, the residual image has no speckles. But it will still have the shot noise of the subtracted speckle patterns—which can be reduced via longer integration time.

Many types of differential imaging have been employed in the various ground-based coronagraphs currently in operation. One of the more common techniques, and one currently baselined by the Roman coronagraph, is called reference differential imaging (RDI).³⁰ It calls for the instrument to point to a “reference” star to generate the dark hole and point back to the reference star every few hours to recalibrate the dark hole (Fig. 15). The final image is the sum of separate reference-subtracted images, in each of which a reference-star image is subtracted from a target-star image (see also Sec. 1 and Appendix A). For the purpose of this paper, we adopt an RDI observing scenario.

Fig. 15

A Roman-like observing scenario for coronagraph applications. A bright reference star serves both as an efficient object for dark-hole creation and to provide a reference speckle pattern for differential imaging.

What differential imaging makes possible is a relaxation of the requirements on direct starlight suppression, if the speckle pattern in the reference image is sufficiently close to that of the target image. Typically, the goal is to achieve an order of magnitude improvement in the suppression using differential imaging. This benefit comes at the cost of a requirement on the stability of the optical system—a requirement that is often the most challenging in a coronagraph, and hence one of the most important tolerances to determine. It is the purpose of this paper to develop a process for determining these tolerances. Note that the stability error budget (Sec. 6) applies to the telescope from the end of the reference integration period to the end of the target integration period. Telescope stability sets the chop period. For a telescope with no instability, there is no need to return to the reference star.

The foregoing discussion, however, should not be interpreted as implying that raw contrast is not important. Understanding the interplay between the existing residual starlight and its change arising from optical instability is important to the analysis that follows. At the end of the wavefront control procedure that gives the coronagraph its final level of starlight suppression, some residual optical error remains, leading to a “leakage” field $E(u,v)$ in the image plane (where $u,v$ are image plane coordinates). For this discussion, it is adequate to think of this as a complex scalar function of position. The speckle intensity pattern is then simply given by ${|E(u,v)|}^2$. When a disturbance or drift error occurs in the optomechanical configuration of the telescope or coronagraph, this field changes by a small amount. We can equivalently think of this as a perturbation field $\mathrm{\Delta }E(u,v,t)$ being coherently added to the “initial” or “static” part of the field $E(u,v)$. The coherent mixing of the original field $E$ and perturbation field $\mathrm{\Delta }E$ creates the modified speckle pattern, the intensity of which is given by

Fig. 16

Example case of mixing fields, with nominal chop (RDI switch) segments, highlighting individual image frames and single-stare integrations. The perturbing field $\mathrm{\Delta }E$, if coherent with the larger existing field $E$, is amplified when mixing with this field. If the perturbation ($\mathrm{\Delta }E$) has multiple oscillations over a long integration time, the mixing term ($2\mathcal{R}\{E^{*}\mathrm{\Delta }E\}$) will average to zero, leaving only the perturbation term ($|\mathrm{\Delta }E{|}^2$). If the perturbation term is slow, the mixing term has a large effect. The perturbation is defined as the change in the field between the reference and measurement.¹⁴

Viewed in terms of filters, observing scenarios can be designed to reduce the effect of the instability terms (usually dominated by the mixing term) by a judicious choice of integration times and chopping. Integration is a low-pass filter, and chopping, which is temporal differentiation, is a high-pass filter. Their combined application can produce a limited band filter to minimize the impact of disturbances.

Another implication of the mixing term is that the amplitude of raw contrast or initial contrast is important in determining speckle amplitudes, so that the assumption of initial contrast cannot be decoupled from setting requirements on optical stability. We will revisit this in Sec. 5 when we discuss the modeling of the modal instability errors.

4.3.

Converting Contrast Instability to Flux Ratio Noise

Earlier, we derived an expression for calculating the flux ratio noise that arises from a source of photometric noise. Having introduced some of the considerations with regard to speckle instability, we now derive the corresponding relationship between speckle instability and flux ratio noise.

In each RDI differential image, there are two stares involved: one at the target star and one at a reference star. There are, correspondingly, two speckle patterns. We can label the two-dimensional contrast maps in the dark hole as $C_{\mathrm{tar}}(u,v)$ and $C_{\mathrm{ref}}(u,v)$, for the target and reference stares, respectively. We will call their difference the residual contrast map $\mathrm{\Delta }C(u,v)$:

Since the peak of the PSF is centered within the region of interest in this case, $\tau (u,v)$ is also referred to as the peak throughput $({\tau }_{\mathrm{pk}})$. The numerator contains the peak throughput ${\tau }_{\mathrm{pk}}$ and $n_{\mathrm{core}}$ the number of diffraction modeling pixels in the image plane covering the PSF core. Though not exactly 1, ${\kappa }_c$ is usually not far from 1. For the vector-vortex charge-6 coronagraph, near the IWA ${\kappa }_c=1.12$.

4.4.

Integration Time Needed to Achieve SNR

The observing scenario, in which the error budget is based, includes the all-important integration time. Time is a key parameter for at least two reasons. First, different types of errors have different time dependencies: random errors can be reduced by integrating longer, while (systematic) drift errors (such as thermal errors or DM actuator drift) grow with time. Second, for a space mission, integration time is a scarce resource, and target selection and the science objectives cannot be decided without counting this cost. Thus the integration time chosen for the observing scenario must be realistic from the standpoint of the random and drift errors. Success in direct detection of a planet can be parameterized in terms of achieved SNR in a given amount of time, or conversely the time required to achieve a desired SNR. This time defines the maximum desired duration for the telescope’s wavefront stability. If sufficient stability within this duration cannot be achieved, then the dark hole will need to be recalibrated. In this section, we develop an analytical expression for the time required to achieve the desired SNR and use the result to calculate the random noise part of the error budget.

To begin, SNR is simply the ratio of the signal $S$ to the noise $N$:

Two important caveats are in order at this point. The first is that the background, which is the result of an RDI differential image taken from images that contained speckles, may not be strictly Gaussian, which means Eq. (21) may be too optimistic.³² The second is that any analytical calculation that estimates $S$ gives the expected value of $S$ and not its distribution. That is, the SNR that we calculate represents (all else being equal) the average SNR from an ensemble of visits. In a typical direct-imaging case, tens of hours must be spent integrating to receive a significant and unambiguous signal. The assessment (partway through a planet-discovery observation) of whether or not a planet has been detected at some location in the dark hole depends on the significance of the excess. Even for the case of a true planet existing in the dark hole, when an integration time suggested by Eq. (21) has elapsed, in half of the cases the SNR will be less than the chosen threshold (because $S$ will be less), and in half of the cases it will be more. To have a high probability that the lack of significance is due not to statistics but from the absence of a true planet, a larger SNR target than suggested by Eq. (21) will be necessary.

In most cases, detection is not the only goal; photometry (i.e., the calculation of the flux ratio of the planet) is also desired. For photometry, the uncertainty in the flux ratio measurement is an important factor. As such, the noise $N$ must now also include the noise associated with the signal. Since the signal $S$ follows Poisson statistics, with the special feature that the mean ($S$) is equal to the variance (${\sigma }^2$), we can then say that the Poisson noise (commonly called shot noise) associated with the signal is simply $\sqrt{S}$. Then $N$ will be the quadrature sum of the two noise contributions, and we get the following for the photometric SNR:

We can re-express this SNR equation in terms of ${\sigma }_{\mathrm{tot}}$ as simply $\mathrm{SNR}=S/{\sigma }_{\mathrm{tot}}$. Furthermore, using Eq. (7), we can break up ${\sigma }_{\mathrm{tot}}$ into a random part (shown below) and a systematic part [which is just ${\sigma }_{\mathrm{\Delta }I}$ in Eq. (7)]. The random variance is given by

In Eq. (28), the subtraction in the denominator causes a divergence in the dependence of the integration time on $\mathrm{SNR}$. If the speckle subtraction is not effective (i.e., $r_{\mathrm{\Delta }I}$ is too large) or the required SNR is too high relative to $r_{\mathrm{pl}}$, the available count rate from the planet, the denominator, can vanish or become negative, indicating no solution. Thus it is useful to define, for a given observation, the critical SNR, ${\mathrm{SNR}}_{\mathrm{crit}}$, where the denominator goes to zero. This is the infinite-time limit of the maximum achievable SNR:

The integration time needed to achieve a desired SNR is architecture-dependent. To calculate the integration time for a specific architecture, we can use the steps in Sec. 2 to compute the peak throughput (${\tau }_{\mathrm{pk}}$), core throughput, PSF size on the sky, contrast, and the number of core modeling pixels. As outlined in Appendix A, the noise, planet, and speckle rates are all calculable based on the observing scenario assumptions for these quantities.

As an example, Fig. 17 plots the time needed to reach an SNR of 7 for a 210-ppt flux ratio exo-Earth. The horizontal axis is $r_{\mathrm{\Delta }I}$, normalized to $r_{\mathrm{pl}}/\mathrm{SNR}$. Per Eq. (28), when this quantity reaches unity, the time to SNR becomes infinite. Larger values of this quantity imply more relaxed requirements on speckle stability and postprocessing effectiveness, but, beyond some point, consuming further integration time to allow more relaxed requirements on speckle suppression has no value. We, therefore, choose an integration time of 25 h for the observing scenario.

Fig. 17

The calculated time to reach SNR as the residual speckle $r_{\mathrm{\Delta }I}$ is increased. In the horizontal axis, $r_{\mathrm{\Delta }I}$ is normalized by $r_{\mathrm{pl}}/\mathrm{SNR}$. The rapid increase of needed integration time as $r_{\mathrm{\Delta }I}$ is increased indicates that there are diminishing returns beyond about 25 h of integration time. We hence choose 25 h as the nominal integration time for the observing scenario.

4.5.

Error Budget at the Top Level

With the target, observing scenario, and error metric all defined, it is now possible to create an error budget. For a 210-ppt exo-Earth target, desired to be observed with $\mathrm{SNR}=7$, the total error from all sources combined must be $<30\mathrm{ppt}$. Using equations derived in Appendix A for the photometric error contributions, Fig. 18 plots the total random noise and its constituents as a function of integration time. Also plotted in Fig. 18 is the maximum allowable systematic error from residual speckle, i.e., the quadrature difference between the total allowable error ${\xi }_{\mathrm{tot}}$ (30 ppt) and the total random error ${\xi }_{\mathrm{rnd}}$:

Fig. 18

The relative contributions to the flux ratio noise from the main noise sources versus integration time. The selected integration time of 25 h is highlighted. Coincidentally, the planet shot noise trace is hidden beneath the detector trace.

A top-level error budget based on these numbers is shown in Fig. 19. The requirement of $\mathrm{SNR}=7$ on the exo-Earth means a maximum allowable flux ratio noise of 30 ppt for the target system of the observing scenario. As shown in Fig. 18, the random error, for a 4-m telescope with a VVC-6, is estimated to be just under 16 ppt, and we use this number for the allocation to random noise. The remainder, in a quadrature sense, goes to residual speckle and reserve. Here we choose 22.5 ppt for the residual speckle error $\delta {\xi }_{\mathrm{\Delta }I}$, which leaves 12 ppt for reserve. Obviously, some degree of freedom exists at this point, but our 12 ppt of reserve amounts to a modest 16% reserve, in a quadrature sense, relative to the total allowed flux ratio noise of 30 ppt. Note that reducing the reserve to zero would only modestly increase the residual speckle allocation to 26 ppt. The last step is to calculate via Eq. (17) the allowed final residual contrast instability after postprocessing. Assuming a residual speckle allocation of 22.5 ppt and a postprocessing suppression factor of 0.5, the allowed residual contrast instability is 40 ppt.

Fig. 19

Top-level error budget for direct imaging of exo-Earth at 10 pc.

The remainder of this paper focuses on how this 40 ppt contrast instability is suballocated to modal instability errors in their native units. The starting point for that process is the coronagraph diffraction model that yields the sensitivities to the various modal errors.

5.1.

How Wavefront Control is Used to Generate the Dark Hole

All high-contrast coronagraphs depend on wavefront control to create the deep null in the dark hole. One widely used technique is the so-called electric field conjugation (EFC) method.²⁸ EFC is an iterative process of measuring then cancelling the electric field in the dark hole. In each iteration, the existing dark-hole electric field is measured through the application of specific surface changes at the DM. These deliberate perturbations, called “pokes,” cause a relatively large amount of additional coherent light $(\mathrm{\Delta }E)$ to enter the dark hole, in a specific pattern, and mix with the existing unknown field. Equation (14) applies here too, but in this context $E$ represents the existing small field leading to the raw contrast, whereas $\mathrm{\Delta }E$ is the intentional, large additional field arising from the probe. The coherent mixing term amplifies the faint initial dark-hole field, giving rise to a now conveniently bright intensity pattern. Ideally, $\mathrm{\Delta }E$ is generated to be flat over the domain of interest. A forward propagation model then guides the determination of the DM shape needed to produce the desired amplifying coherent field at the dark hole. Pairwise, complementary real and imaginary amplification fields in the dark hole, generated via the appropriate poke patterns on the DM, allow the estimation of the dark-hole field.³³ Once the (preamplification) dark-hole field is estimated, the DM is restored to its unpoked state. In the next step, the new DM shape, that would cancel the measured dark-hole field, is estimated. The estimation process uses a stored library of sensitivities obtained using the forward diffraction model. These are sensitivities of the real and imaginary parts of the electric field at each dark-hole pixel to a perturbation in each DM actuator. A pseudoinverse solution, with constraints added to limit actuator strokes in a single iteration, is used to compute the shape. Errors in knowledge of the optical configuration, the DM actuators’ response to commands, the detector noise, as well as the first-order approximation implicit in the EFC method, all contribute to error in the estimated shape. Therefore, it is necessary to repeat the correction process many times to achieve the final contrast—i.e., “dig” the dark hole. Whether the desired contrast is attained in an acceptable number of iterations, or not at all, depends to a large measure on the accuracy of the system model, the DM and the detector noise.

DM actuator commands cannot perfectly compensate for the imperfections of the DM surface. To achieve a ${10}^{-10}$ level of raw contrast, the DM must produce a wavefront that deviates by no more than order of (10 pm) from ideal. But real DMs never reach anywhere near this level of perfection. If a DM was commanded to a flat surface, the residual surface error would never be better than a few nanometers rms. This apparently insurmountable obstacle can be circumvented because many more solutions than the assumed ideal wavefront exist. A gap of nearly three orders of magnitude between the flatness requirement and what can be achieved is closed using a wavefront control algorithm that searches for a local “optimum” solution near the current surface. It makes minute, subnanometer adjustments to the DM’s current shape, in the vicinity of the surface shape the DM has already reached (with its nanometers of error). Put another way, the final iterations of wavefront control do not necessarily improve the root-mean-square surface error, but they do move light from the inside to the outside of the dark hole. Some coronagraph designs, like the hybrid Lyot, require a specific nonflat initial wavefront error to be applied by the DM. In this case, a perfect stellar wavefront enters the telescope, the instrument optics add error to this flat wavefront, and the DM further adds a design-specified pattern to the wavefront. The DM’s shape must both compensate for the wavefront error of the optics and add the needed wavefront shape required by the coronagraph. For other coronagraph designs, the coronagraph needs a flat wavefront. In these cases, the DM merely compensates for the wavefront error incurred from reflections and transmissions through the optics.

5.2.

Wavefront Error Approximation

With these considerations in mind, the approach employed in this paper involves generating the initial raw contrast by the addition of a small wavefront error (a few pm rms) to the coronagraph’s per-design ideal wavefront. For vector-vortex and apodized pupil coronagraphs, the ideal wavefront is flat, while for the hybrid Lyot it is a specific nonflat shape. This approach is summarized in Fig. 20.

Fig. 20

Logical steps to the guiding assumption on wavefront error simulation. In this example, it is assumed that the ideal wavefront in this coronagraph design is a flat wavefront. This would be true for the typical apodized pupil or vector-vortex case.

As Fig. 20 illustrates, the coronagraph masks are designed for an ideal (e.g., flat) incoming wavefront. Real DMs, as noted above, can only reach a desired wavefront with the fidelity of a few nanometers rms. This is because their phase sheets have deformations at spatial frequencies above the actuator spacing and because the actuators themselves can produce residual phase-sheet deformations. Obviously, nanometers of error are far from ideal for reaching the required level of contrast. To overcome this limitation and dig the dark hole, wavefront control makes small, pm-level adjustments to the DM to find a high-contrast solution near the best-effort attained shape. Finally, from a modeling perspective, the important point (illustrated as step 4 in Fig. 20) is that the (actuator height) phase-space distance between the optimal local solution and the EFC-settled solution can just as well be applied to a simple-to-simulate, ideal surface (e.g., the ideal flat surface for a vector-vortex coronagraph).

For the purposes of this paper, the DM error spectrum is assumed to be flat for all spatial frequencies out to the edge of its sampling. For example, a DM with 64 actuators across its surface is assumed to have a surface error power spectrum that stays flat out to the edge of its Nyquist frequency of 32 cycles per aperture then rises abruptly beyond this range. An azimuthally integrated PSD distribution and the resulting surface shape for such a DM is modeled in Fig. 21. Note that this is not meant to represent a real DM surface but a DM surface error relative to the final wavefront-control computed surface. This error (in both a real system and here in the model) is the dominant contribution to the raw contrast.

Fig. 21

Wavefront error implementation following the approach outlined in this section. (a) The assumed azimuthally integrated PSD, which is flat and small out to 32 cycles per aperture (where a $64\times 64$ DM is expected to have strong control authority) and rises abruptly afterward, falling off with spatial frequency as $f^{-2.5}$. (b) The resulting surface shape.

The procedure outlined is relatively straightforward to implement, but it does not predict the initial contrast that can be expected. Instead, we assume that a certain level of contrast has been achieved via wavefront control, and then we evaluate sensitivities to various modes of wavefront error in the presence of an initial field consistent with the assumed contrast. The Roman mission technology effort has demonstrated contrast approaching ${10}^{-9}$ in testbeds.³⁴ It is reasonable to expect that a future telescope, customized for a specific coronagraph, could achieve another order of magnitude better performance in contrast. Therefore, we assume a range of initial contrast values in the ${10}^{-10}$ vicinity and compute a sensitivity curve for each wavefront error mode.

5.3.

Contrast Instability and Residual Speckle

After all measures to create the deepest null have been taken, the last step in improving signal detection is to calibrate and subtract the remaining speckle pattern. We assume RDI is the method used. For the reference image to remain accurate, the speckle pattern needs to remain stable. When speckles are perfectly stable, RDI yields a residual that only has random (photon and detector) noise. But if the speckles have instability, RDI yields a residual with “structure.” As we noted already, the residual speckle standard deviation ${\sigma }_{\mathrm{\Delta }C}$ depends on raw contrast. But, in modeling, setting the raw contrast to a precise value is not convenient. It is easier to start with an assumed input field and propagate that field forward to arrive at a contrast map. Repeating this exercise with increasing amplitudes of effective post-EFC surface error (Fig. 21) produces a set of dark holes with increasing amounts of initial raw contrast. The coronagraph error budget in general will call for a specific initial raw contrast. That particular value might not exactly correspond to any of the raw contrast values we have created, but, as long as our range straddles the desired contrast, we can interpolate our results to the called-for initial raw contrast.

To get the sensitivity of the contrast to specific wavefront perturbation modes, we start with some initial field and add the perturbing field. The initial field creates the initial raw contrast in the dark hole, and the perturbed field causes the changed contrast. Subtracting the perturbation-added speckle map from the original speckle map gives the residual speckle map. We are interested in the structure of the residual contrast map as quantified by its SSD within the dark hole. Since the contrast, and residual contrast, change most strongly with angular separation (i.e., radius), it is usual to divide the dark hole into radial slices and compute the standard deviation within each radial slice. The sensitivity is the ratio of this quantity over strength of the perturbation:

Table 1

List of error modes considered.

Note that monolithic and segmented aperture telescopes have different perturbation modes. For example, regardless of whether the telescope has a monolithic or segmented primary mirror, physical displacements of the secondary mirror relative to the primary mirror produce global tip/tilt, defocus, astigmatism, coma, and spherical errors. But only a segmented aperture mirror has segment-to-segment piston and tip/tilt error, and only a segmented aperture mirror is affected by backplane bending because the individual mirror segments are mounted to the backplane.

For the global modes, we assume that the backplane has the shape corresponding to the mode considered (such as a global bend). We then compute the piston and tip/tilt of a tangent plane centered at each segment’s two-dimensional center and apply the piston to the segment at its center. We also apply the local tip/tilt arising from the global deformation. For the azimuthally symmetric cases, such as defocus, we create only one instance, while for the rest, we randomly create a number of instances with different azimuthal orientations (clocking). We then average the results for the set.

Figure 22 shows an example of this procedure for the case of a 4-m off-axis unobscured monolithic aperture telescope with a vector-vortex charge-6 coronagraph. The left image shows the residual speckle map $\mathrm{\Delta }\mathrm{C}$ when the wavefront is changed by 10 pm of trefoil. The residual contrast map can be divided into annular zones. For each zone, the azimuthal standard deviation (${\sigma }_{\mathrm{\Delta }C}$) of the residual speckle map is computed. The contour plot in Fig. 22 shows how ${\sigma }_{\mathrm{\Delta }C}$ varies with angular distance in $\lambda /D$ and perturbation amplitude. If we set a requirement that the residual speckle contribution from trefoil shall not exceed ${10}^{-11}$, then the region excluded from observation is that to the left of the—11 contour line (shaded). And, if the goal is to observe an exo-Earth located at $3.6\lambda /D$ (yellow vertical zone), then this graph shows that the trefoil change between reference and target star observation times must be less than $\sim 3\mathrm{pm}$ rms. Repeating this process for each perturbation mode fully populates the telescope/coronagraph sensitivity table.

Fig. 22

(a) Residual speckle and (b) contours of ${\sigma }_{\mathrm{\Delta }C}$ for different levels of perturbation wavefront error amplitude and radial slice. (a) A $1\lambda /D$-wide annular region at this separation is highlighted by a pair of concentric circles. (b) The contour values are of ${\log }_{10}{\sigma }_{\mathrm{\Delta }C}$. These plots correspond to the case of trefoil mode on vector-vortex charge-6 coronagraph. The vertical yellow strip is centered at the separation of our nominal target (exo-Earth at 10 pc) observed at a waveband centered at 550 nm.

8.1.

Differential Image

We would like to know the contribution of the background to the noise in the differential image. We assume there is a bright (typically three magnitudes brighter) reference star available, and the telescope slews periodically to this star and obtains reference speckles for subtraction. The total time for obtaining the measurement, excluding the slew and settling times, will be $t_{\mathrm{tot}}=t+t_r$, where $t$ is the integration time on the target star, and $t_r$ is the integration time on the reference star.

We can think of the target star image as a two-dimensional map (counts per unit area), which we can write as $I_{\mathrm{tar}}(u,v)$. Dropping the explicit notation of the coordinates from now on, for the reference star, we have a corresponding quantity $I_{\mathrm{ref}}$. The differential image is obtained by subtracting the scaled reference image from the target image

In the absence of noise and any planets, and if the reference star speckle pattern is identical to the target star case, then $\mathrm{\Delta }I(x,y)$ should be zero everywhere.

The sources of random noise are:

The first three error sources are photonic in origin and require a discussion of throughput. We have neglected stray light in this list, assuming good rejection of this background. If it is included, the treatment might begin as being most similar to local zodi (smooth incoherent background), for example in the calculation of the RDI noise enhancement factor discussed below.

The main source of nonrandom error is the residual speckle after differential imaging. This, which we will label as systematic error, is much more difficult to model analytically, but we will treat this with a simple parameterization further below.

8.2.

Throughput

The throughput can be expressed as a product of factors.

In Sec. 2, we defined core throughput. In terms of these factors, core throughput is given by

These considerations imply

8.3.

Planet Signal

The signal, which appears only when observing the target star, has an expected value of

8.4.

Total Noise Variance

The total noise variance can be written as

8.5.

Random Noise Variance

We distinguish the contributions to ${\sigma }_{\mathrm{rnd}}^2$ from the target and reference star observations:

8.6.

Speckle Noise Variance

The second term of Eq. (44) ${\sigma }_{\mathrm{sp}}^2$ is the variance from the shot noise in the starlight halo (i.e., the mean speckle background). The first step is to calculate the mean speckle rate $r_{\mathrm{sp}}$. We express this, following the discussion in the main narrative that led to Eq. (3), in terms of the fraction of the starlight that ends up in the region of interest. We assume that the region of interest corresponds to the PSF core region, with size ${\mathrm{\Omega }}_{\text{core}}$. Per Eq. (3), the fraction of light diffracting into ${\mathrm{\Omega }}_{\text{core}}$ is given by

8.7.

Zodi Noise Variance

The third contribution is the noise from zodiacal background ${\sigma }_{\mathrm{zo}}^2=r_{\mathrm{zo}}t$. There are contributions both from our own solar system (“local-zodi,” lz) and the exo-system (“exo-zodi,” ez). Since these are extended sources of light, they are expressed in terms of spectral flux per unit solid angle. Thus the zodi count rate is given by

Actually, the zodi variation over the dark hole may not be uniform for the exosystem. Here for simplicity, we have assumed it is uniform with the same effective throughput ${\tau }_{\mathrm{zo}}$ as local zodi. This is a conservative assumption, as it allows more throughput for this very significant background. At present, statistics on exosystem dust densities is very poor, so a conservative estimate is reasonable.

8.8.

Detector Noise Variance

The detector used in the Roman coronagraph is an EMCCD, with very low effective read noise.³⁶ The HabEx concept mission has also baselined a similar detector. The EMCCD detector noise variance is given by

8.9.

Additional Noise from Differential Imaging

So far, we have only considered the random noise from target star observations. But the differential image is also affected by the noise from the reference image, albeit smaller because the reference star is brighter. Here, we account for the additional noise coming from the reference star.

We will first examine the speckle background shot noise within the region of interest—the core region centered on a putative planet. Since the intensity is proportional to detected (electron) counts, we can restate Eq. (36) as $\mathrm{\Delta }n(u,v)=n_{\mathrm{tar}}-\beta n_{\mathrm{ref}}$. Since we are not including the planet, we should have $\mathrm{\Delta }n\approx 0$, so that the normalized reference counts are equal to the target counts, or $\beta n_{\mathrm{ref}}=n_{\mathrm{tar}}$. We write the noise in the differential image as ${\sigma }_{\mathrm{tot}}^2={\sigma }_{\mathrm{tar}}^2+{\sigma }_{\mathrm{ref}}^2$. From Poisson statistics, ${\sigma }_{\mathrm{tar}}=\sqrt{n_{\mathrm{tar}}}$. The reference star contribution, because of the normalization factor, is ${\sigma }_{\mathrm{ref}}=\beta \sqrt{n_{\mathrm{ref}}}$. But, since $\beta n_{\mathrm{ref}}=n_{\mathrm{tar}}$, the net result for the additional noise from the reference star to the speckle ($\mathrm{sp}$) noise is

The detector clock-induced charge contribution from the reference star is related to the contribution from the target star in the same way as the dark current, by simple inspection of the form of these two error sources in Eq. (48). Thus when the normalization factor from differential imaging is included, the net effect is that ${\sigma }_{\mathrm{ref}}=\beta \sqrt{t_r/t}·{\sigma }_{\mathrm{tar}}$ for detector clock-induced charge.

Since the two detector terms are related the same way, so is their quadrature sum. Thus, overall, the detector (det) part of the reference noise is related to the target noise according to:

The factors are summarized in Table 2 for the five error categories that apply. There are two ways one can bookkeep the effect of differential imaging: either add up all the reference star contributions in quadrature into one total number, to be added in quadrature itself with the target star contribution, or apply an error enhancement factor to each of the error sources to account for the differencing. The bottom row in Table 2 gives the variance enhancement factors $k_x$ for each error source. For the speckle, for example, this table is to be read to give $k_{\mathrm{sp}}=1+\beta$, etc.

Table 2

Reference star error relative to the target star error contribution for each of five categories that apply to HabEx operation under an RDI observing scenario.

We can now define, following Eq. (44), a total random noise variance rate $r_n$, which includes both contributions:

8.10.

Modifications to Accommodate Spectroscopy

We have so far been assuming a direct imaging scenario. The HabEx coronagraph will also be using an IFS for spectroscopy. To extend the results to spectroscopy, we need to make an abstraction.

We can think of spectroscopy as photometry in the separate spectral elements of a spectrum. For a concrete example, the HabEx concept mission plans to do spectroscopy with a spectral resolution of $R_{\mathrm{IFS}}=\lambda /\delta \lambda =140$, where $\delta \lambda$ is the spectral width of a single spectral element.³⁷ A filter of bandwidth $\mathrm{BW}=\mathrm{\Delta }\lambda /\lambda$ is used in each coronagraph observation. The instantaneous bandwidth of HabEx is planned to be 20%.

The SNR of interest for spectroscopy is for a single spectral element. That is, we have the light in the PSF core now subdivided spectrally into $N_{\mathrm{spec}}$ spectral elements:

8.11.

Nonrandom, Systematic Background

We now turn to the systematic contribution to the total noise variance ${\sigma }_{\mathrm{\Delta }I}^2$, the second term in Eq. (42). This is the noise from the nonuniformity of the residual speckle in the dark hole, so that a planet can be confused for one of these, or vice versa. Focusing only on the speckle background, the differential image is given by

The measure of nonuniformity is the SSD of the differential image excluding known planets. Since all the diffractive effects are band-limited to the $\lambda /D$ scale, no features finer than scale (such as detector noise, which produces effects on the pixel scale) can be optical. Hence, in computing the SSD, it is acceptable to low-pass filter the image (e.g., average the pixels) down to the $\lambda /D$ scale before computing the SSD. The domain of interest is where a signal could be the entire dark hole, or more typically a radial slice within the dark hole:

8.12.

Derivation of the Relation between Contrast Stability and Flux Ratio Noise

The systematic noise ${\sigma }_{\mathrm{\Delta }I}$ is defined as the variance, on the core-area scale, of the postsubtraction residual (e.g., RDI). The core area on the sky is roughly $\lambda /D$ in width. The temporal characteristics of ${\sigma }_{\mathrm{\Delta }I}$ depend on the type of perturbation. For oscillatory variations in the speckle, such as might arise from dynamic modes of the telescope, the time constant would probably be shorter than the integration time $\tau \ll t$ and thus ${\sigma }_{\mathrm{\Delta }I}$ would decrease rapidly with increasing integration time. However, for long-time constant error sources, such as actuator drift or the slowest thermal variations, the time constant would be large relative to the integration time $\tau \gg t$ so that the variance ${\sigma }_{\mathrm{\Delta }I}^2$ would grow at the same rate as the signal (i.e., ${\sigma }_{\mathrm{\Delta }I}^2\propto t^2$). Since this mode of temporal dependence dominates over the other, we will use a temporal parameterization that only includes this mode. So, we define a postsubtraction improvement factor relative to the mean speckle rate $r_{\mathrm{sp}}$:

In error budgeting, $f_{\mathrm{\Delta }C}$ is usually informed by separate integrated modeling results where the effects of various thermal and jitter errors are propagated to the final image. Similarly, the choice of a good factor to use for $f_{\mathrm{pp}}$ is informed by postprocessing of model-generated speckle histories.

Thus for the systematic contribution ${\sigma }_{\mathrm{\Delta }C}$, we have, by the definition above, the explicit integration-time dependence:

It is useful to define a differential image error rate $r_{\mathrm{\Delta }I}$, which is the expected rate at which the differential image SSD grows:

8.13.

RDI Scenario SNR Equation

Putting all that we have now defined together, we rewrite the expression for the photometric SNR, $R$, given by Eq. (22), to make the dependence on integration time $t$ explicit:

8.14.

Time to Reach SNR

We can ask how long we have to integrate on the target star before we reach a desired SNR, $R$, given the error rates, including the enhancement factors. Solving Eq. (74) for the integration time $t$, we have