In the last two decades, the existence of 3,498 exoplanets has been confirmed,1 and in the upcoming era of 30-m class ground-based telescopes and new space-based observatories, there is the promise of discovery, even characterization, of many more exoplanets, including potentially Earth-like worlds. With such powerful capabilities on the horizon, it has become imperative to push stellar suppression technology to higher precision in order to directly image these exoplanets. Combined with coronagraphy, current speckle nulling techniques2 are capable of creating the dark field (DF) in the science image where light from an exoplanet orbiting its star can be detected. In principle, these methods are capable of generating a DF with contrast, which would enable the imaging of Earth-like exoplanets,3 but to generate and maintain a DF at such contrast requires high-precision measurements of very small changes in the science image.
The DF is very susceptible to small dynamic aberrations in the beam path. Aberrations produce postcoronagraph stellar light leakage and result in a quasi-static speckle field in the image plane that limits the DF contrast.4 Conventional wavefront sensors (WFS) that branch off from the main beam path cannot correct for noncommon path errors, and the resulting quasistatic aberrations can eventually become larger than the residual dynamic wavefront errors after correction by the WFS control loop. For high-contrast imaging, the WFS should ideally be common path with all of the optics seen by the science instrument to allow access to all aberrations created in the science beam. This is achieved by focal plane wavefront sensing (FPWFS), which uses the science detector as the WFS to measure the exact aberrations seen by the main science beam.5,6 FPWFS techniques like speckle nulling and electric field conjugation (EFC) that have proven capable of generating a DF with high contrast in the lab have also been under consideration for maintenance of the DF.7–9 As a control method, FPWFS presents its own set of challenges given that speckles have a quadratic relationship with aberrations. These techniques also rely on phase diversity measurements of the field at the science detector, which require field modulation and multiple images.10,11 This field modulation at the science detector, induced by a deformable mirror (DM), throws stellar light back into the DF and disrupts the science measurement. This interruption, which is required to rebuild the DF every time the contrast degrades, fundamentally limits the integration time that can be spent on any given target. The duration of this interruption to the science acquisition is directly related to the contrast of the DF. For deeper contrast, the required exposure time to sense the speckle field increases; therefore, at the contrast level, multiple images with long exposure times significantly reduce the amount of time that can be spent on observations. The need for modulation, multiple images, and long exposures, consequently makes the use of current speckle nulling methods and EFC nonideal for continuous maintenance of the DF.
Another technique known as the self-coherent camera (SCC) has been under development as a method for obtaining and maintaining the DF without science acquisition competition.6 While SCC does not require modulation, it does still utilize the mixing of some starlight with the DF. Linear dark field control (LDFC) does not require any such mixing of starlight with the DF and offers a potential solution for overcoming the limitations presented by speckle nulling and EFC. To avoid disrupting the science measurement with field modulation to rebuild the DF, LDFC locks the high-contrast state of the field once the DF has been constructed using conventional methods like EFC. Using only one image of the bright field (BF), LDFC freezes the state of the field by sensing and canceling changes in the wavefront that result in speckle formation in the image plane. The ability to maintain the DF with a single image yields a substantial increase in time that can be spent in the observation and analysis of exoplanets and will lead to an overall increase in the number of planets detected and analyzed over the lifetime of an instrument.
LDFC maintains high contrast without modulating the field to update the field estimate as is required when using EFC in closed loop. Conceptually, LDFC is inspired by low-order wavefront sensing (LOWFS), a proven technique designed to actively sense and correct low-order wavefront aberrations using light rejected by a reflective stop placed in a pupil plane12 or focal plane13,14 within the coronagraph. Instead of sensing only low-order aberrations using starlight rejected by the coronagraph, LDFC also has access to mid- and high-spatial frequencies. It is a FPWFS that operates a closed loop around starlight outside of the DF.
Spatial LDFC freezes the state of the DF by using state measurements of light spatially outside of the DF (see Fig. 1). This method uses the linear signal from the strongly illuminated BF to measure the change in the image plane intensity and uses that variation to calculate the correction required to return the image to its initial state, thereby stabilizing the DF. Unlike other FPWFS techniques, spatial LDFC does not rely on any induced modulation to derive an estimate of the field to be canceled. Instead, spatial LDFC observes the changes in the image intensity with respect to a reference image taken after the DF has been established by conventional speckle nulling methods. This process requires only the reference image and a single image taken at a later time.
Without the need for modulation or multiple images, spatial LDFC does not interrupt the science measurement, it decreases the time necessary to return the DF to its initial high-contrast state, and consequently it allows for longer, uninterrupted observing at high contrast. This paper introduces spatial LDFC as a more efficient alternative to conventional speckle nulling methods for stabilizing the DF. The theory behind spatial LDFC is laid out here in Sec. 2, and demonstrations of LDFC’s abilities in simulation are shown in Sec. 3. Further discussion of the limitations and null space of spatial LDFC is laid-out in Sec. 4 with concluding remarks in Sec. 5.
Spatial LDFC relies on the linear response of the BF to wavefront perturbations that affect both the BF and the DF; this linearity allows for a closed loop control algorithm directly relating wavefront perturbations to changes in BF intensity. To derive the source of this linear response, we begin with the relationship between an incident wavefront and the resulting image. The complex amplitude of the incident wavefront in a pupil plane is linearly related to the complex amplitude at the image plane at a given time . The same linear relationship is true with respect to , the multiplicative complex amplitude introduced by the DM in a conjugate pupil plane, and the complex amplitude at the image plane .
When the changes in optical path length (OPL) induced by the DM are very small such that , the resulting field at a given time in the image plane can be written as the sum of the initial pupil plane field and the small changes in complex amplitude induced in a conjugate pupil plane by the DM10
The resulting intensity in the image plane at time is then given by
The total image plane intensity can be written as a sum of three terms: the intensity contribution from the initial pupil field: , the resulting intensity due to phase perturbations induced by the DM: , and the inner product of the initial pupil field and the DM contribution to the complex amplitude
The intensity of the BF at the image plane at time can, therefore, be written as a linear function of the complex amplitude contribution of the DMFig. 2 alongside the quadratic response of the DF to the same DM perturbation. In this figure, the BF and DF response to the DM field contribution is shown in a simulated PSF with a DF established by conventional EFC. A model of a MEMS DM was used to create the DF and then perturb the input wavefront by inducing a positive and negative delay in the optical path with a single actuator. This was done for a range of actuator amplitudes from to .15 The resulting intensity response of pixels located in the DF [shown in Fig. 2(a)] is governed by Eq. (3) with the expected quadratic dependence on the field perturbation. The intensity response of the BF [shown in Fig. 2(b)] reveals the predicted linear dependence on the DM-induced field perturbation given by Eq. (6).
In closed loop, is small, and the linear approximation holds. However, even when initially closing the loop where is larger, strict linearity is not required, only a monotonic trend. In instances both of strict linearity or monotonicity, the BF response allows for the construction of a linear servo driven solely by changes in the BF intensity. Unlike EFC and speckle nulling, which use modulation to provide an absolute field measurement, LDFC relies on relative measurements of BF intensity variation in the science image. These intensity variations are used to track and cancel changes in the wavefront that modify both the BF and DF, thereby stabilizing the DF without any disruptions to the science measurement.
Given the linear relationship between BF intensity and wavefront, using LDFC to stabilize the DF contrast is faster and more robust than using EFC. EFC requires multiple images to estimate the field, whereas each iteration of LDFC requires only one image at the science detector to determine how the field has changed with respect to the initial EFC-derived state. Since this image does not require field probing, which breaks the science measurement, the LDFC servo operates with a 100% duty cycle. Furthermore, LDFC does not rely on complex field estimates that require a model-based complex phase response matrix that is difficult to measure and verify; instead, LDFC relies only on a calibration that links a set of DM shapes, or basis functions, to changes in intensity in the science image.16
For this simulation, the DM influence functions were chosen as the basis functions. The calibration between the image and the basis set is obtained by building a response matrix whose columns relate the application of each individual influence function to the responding intensity variation at the science detector. Though modal control17 does offer performance benefits, especially when it maps with the expected temporal evolution of the wavefront error, such modal control tuning has not been explored at this time. For this work, application of the influence function basis set involved the actuation or “poking” of one of the actuators on the DM that lie within the illuminated system pupil. To begin building , the ideal reference image with dimensions  is recorded after the DF has been established using EFC. To fill each of the columns in , a single actuator is poked, the resulting perturbed image is measured, and the unperturbed reference image is subtracted off to yield the change in intensity. This is done for all actuators. The resulting response matrix has the dimensions Fig. 3 for a contrast threshold of .
To build the BF response matrix , the full response matrix is filtered to include only the BF pixels with intensities above the threshold such that . This filtered response matrix is used throughout the operation of spatial LDFC.
To implement LDFC in closed-loop, an image is taken at time and the same BF pixels that pass the threshold are recorded in the BF image . The BF reference image is then subtracted from the new BF image to track the changes that occurred in the BF with respect to the initial EFC BF reference (see Fig. 4)Fig. 5.
In the following simulations, the response matrix and subsequent control matrix were measured once and applied in closed loop with an initial gain of 0.6. Once the DF contrast converged to , the gain was lowered to 0.1 to maintain the correction. The ensuing process of taking an image, calculating the intensity change of the BF from its reference state, and updating the DM was iterated on to actively freeze the science image field in its initial EFC state.
To demonstrate spatial LDFC’s ability to maintain the high-contrast DF, a 6.5-m telescope system was constructed in simulation, which included a single DM and Lyot coronagraph that removed approximately two orders of magnitudes of stellar light from the final image. The system entrance pupil was a circular 6.5-m diameter, centrally obscured mask with a 30% central obscuration, and 2% spiders [see Fig. 6(a)]. The Lyot coronagraph consisted of a Lyot stop undersized by 1% and a focal plane mask with a diameter of . For the system’s DM, a model of a Boston Micromachines 1K DM was defined using 1024 actuators sharing a common Gaussian influence function and 15% interactuator coupling. The diameter of the illuminated pupil projected onto the DM was 6.5 mm, covering actuators and lending an outer working angle (OWA), or control radius of . Sampling at the science detector was per pixel. The source was a magnitude 5 star with sensing done at (V band) with 10% bandwidth. The total flux at the entrance pupil was , and this rate was used to embed photon noise in all of the images in Eq. (8). All of these test parameters are listed in Table 1.
Simulated system parameters used in the following spatial LDFC demonstrations.
To build the DF, a standard implementation of EFC11 was used to suppress the stellar light to an average contrast floor of within a region centered at from the PSF core [shown in Fig. 6(b)]. This DF was the ideal reference state for LDFC to maintain, and the intensity image was saved as the reference image to be used in the LDFC servo to return the DF to its EFC-derived state.
With the DF established, the spatial LDFC algorithm was implemented as described in Sec. 2.3 to maintain the DF in the presence of two separate injected phase aberrations. In the first case, a single speckle pair was induced in the image plane by applying a sine wave phase perturbation in the pupil. For the second case, a random Kolmogorov phase screen was introduced in the pupil creating multiple speckles in the image plane. In both cases, the same optical system, source, and threshold values were kept constant as were all other simulation parameters. The following sections present spatial LDFC’s response to these two cases.
Sine Wave Phase Perturbation
After the DF was constructed, a spatial sine wave phase perturbation with 6 cycles/aperture was introduced into the pupil plane, forming a speckle at : one speckle within the DF and one speckle within the BF. The sine perturbation was given a 1-nm peak-to-valley (P-V) amplitude in phase, creating a speckle pair with a maximum magnitude of and an average aberrated DF contrast of . The LDFC control loop was run with a gain of 0.6 until the average DF contrast reached , at which point the gain was reduced to 0.1 to maintain the correction. The LDFC control loop was allowed to run for 50 iterations for this demonstration with convergence occurring after six iterations. The results are shown in Table 2 and in Figs. 7Fig. 8Fig. 9–10. It should be noted that, in Figs. 9 and 10(b), the LDFC-corrected DF contrast occasionally drops below the initial EFC contrast level. This effect is due to noise fluctuations.
Performance with a sine wave phase: initial EFC DF average contrast, magnitude of the injected speckle, average contrast of the aberrated DF, average contrast of the DF after LDFC, total change in contrast for one full LDFC loop, and the number of iterations to converge to the EFC contrast floor.
Kolmogorov Phase Perturbation
In the first case, the injected aberration created a single speckle in the DF and a corresponding speckle in the BF. To demonstrate spatial LDFC’s ability to suppress multiple speckles, a Kolmogorov phase aberration was generated in the pupil plane instead of a sinusoidal phase perturbation (see Fig. 11). The phase perturbation was given a P-V amplitude of 20.5 nm, creating an aberrated DF with an average contrast of . The LDFC control loop was again run with a gain of 0.6 until the average DF contrast reached , at which point the gain was reduced to 0.1 to maintain the correction. The LDFC control loop was allowed to run for 50 iterations for this demonstration with convergence occurring after six iterations. The results are shown in Table 3 and shown in Figs. 11Fig. 12Fig. 13–14. As in the previous single speckle demonstration, the LDFC-corrected DF contrast occasionally drops below the initial EFC contrast level in Fig. 14(b). This effect is due to noise fluctuations.
Performance with Kolmogorov phase: initial EFC DF average contrast, magnitude of the injected speckle, average contrast of the aberrated DF, average contrast of the DF after LDFC, total change in contrast for one full LDFC loop, and the number of iterations to converge to the EFC contrast floor.
We have demonstrated here that spatial LDFC is capable of locking the DF contrast at its ideal EFC state using only the BF response to a perturbation in the optical path. However, there are limitations to spatial LDFC and a potential null space, which need to be explored. These issues and some potential solutions are addressed below.
Limitations of Spatial Linear Dark Field Control
One significant limiting factor for spatial LDFC is DF symmetry. This technique requires access to a BF that is located spatially opposite the DF. Due to this requirement, spatial LDFC is expected to work only with a nonsymmetric DF. However, in the case of a symmetric DF, spectral LDFC offers a possible solution (see Sec. 4.2). Since spectral LDFC relies on speckles that are located spatially within the DF but outside of the control bandwidth, it is not affected by the lack of a BF spatially opposite the DF. In the case of a much larger DF than the one presented here, spatial LDFC is predicted to still be capable of stabilizing the DF, but it cannot use BF speckles at spatial frequencies higher than those present in the DF to do so.
In future work, spatial LDFC’s null space must also be further explored. For this technique, the null space will consist of wavefront errors that affect the DF without changing the BF. One potential example of this null space is the formation of a speckle on a single side of the focal plane due to the combination of phase and amplitude sine wave aberrations. In such a case, if the speckle falls inside the DF, the BF will not see any modulation and will, therefore, be unable to sense and correct the aberration. A second potential null space example would consist of an incident phase aberration sine wave with a phase that creates a BF speckle with a phase that is 90 deg from the local BF phase. This case would not create a linear signal and would, therefore, not be corrected by LDFC. To date, the simulations presented here have not revealed a null space, but it can exist under system-specific conditions. It should also be noted that this paper has specifically explored a system in which aberrations were introduced and corrected by the same DM; in real systems, there will be aberrations that occur outside the DM-conjugate pupil plane and subsequently do not correspond exactly to DM authority. Such cases will require further investigation as spatial LDFC continues to develop.
Addressing the Null Space with Spectral Linear Dark Field Control
A potential solution for overcoming spatial LDFC’s null space is to operate a separate version of LDFC simultaneously. This second version, known as spectral LDFC, freezes the state of the DF within the control bandwidth by using state measurements of light outside of the control bandwidth. This method exploits the fixed wavelength relationships that exist between speckles at different wavelengths that were generated by the same aberration. To first order, this fixed relationship scales the speckle separation linearly with wavelength and scales the complex amplitude inversely with wavelength. The complex amplitude speckle field may also interfere with static chromatic coronagraph residuals due to the coronagraph’s finite design bandwidth. These relationships between out-of-band and in-band light allow for the state of the DF within the control band to be monitored and maintained by measurements made of speckles located outside of the spectral control band.18 Since spectral and spatial LDFC rely on a BF signal from separate dimensions, the null spaces of the two forms of LDFC are not expected to overlap. For this reason, concurrent operation of spectral and spatial LDFC can provide a powerful tool for compensating for the separate null spaces of both techniques.
As an example of the BF signal that can be used by spectral LDFC, Fig. 15 shows the DF created using a phase induced amplitude apodization (PIAA)19 coronagraph at JPL’s high-contrast imaging testbed (HCIT). Speckles within the DF are shown at multiple wavelengths, both in-band (the science image) and out-of-band (the signal used by spectral LDFC).
While spectral LDFC was not in operation when this image was taken, this is a clear demonstration of a case in which the in-band DF contrast could be maintained by sensing the speckles that are outside the control bandwidth and applying the appropriate wavelength-scaled correction to cancel the in-band speckles. Further development and analysis of this form of LDFC can be found in an upcoming paper by Guyon et al.18
In summary, spatial LDFC acts as an extension of EFC by operating as a servo that can maintain high contrast in the DF during science exposures. Using changes in the BF to provide updates on the state of the field within the DF, spatial LDFC is able to lock the state of the DF after it is established by EFC without relying on field modulation which interrupts the science acquisition and fundamentally limits the exposure time. The substantial increase in uninterrupted observation time spatial LDFC offers makes it a more efficient method than EFC for maintaining deep contrast and will lead to an overall increase in the number of planets detected and analyzed over the lifetime of an instrument. Here, we have introduced the mathematical principles behind spatial LDFC and provided demonstrations of its capabilities through numerical simulation; future work will include further analysis of this technique’s abilities as well as laboratory demonstrations.
This work was funded by the NASA Exoplanet Exploration Program Segmented Coronagraph Design and Analysis study effort. The authors would like to acknowledge Dr. Sandrine Thomas for her assistance in implementing EFC for this work. For their continued support, the authors also acknowledge fellow members of the OG and JRM research group: Justin Knight, Jennifer Lumbres, Alexander Rodack, and Lauren Schatz.
Kelsey Miller is a PhD student at the University of Arizona, the College of Optical Sciences. She has a bachelor’s degree in physics with a focus in astrophysics from the New Mexico Institute of Mining and Technology and a master’s degree in optical sciences from the University of Arizona. Her doctoral research is in high-contrast exoplanet imaging with a focus on focal plane wavefront sensing and control systems for extreme adaptive optics systems.