Implicit Electric Field Conjugation Through a Single-mode Fiber

Detection and characterization of Earth-like planets in the habitable zone is challenging due to the small angular separation and high contrast between exoplanets and their host stars. High contrast imaging (HCI) systems aim to suppress diffracted starlight, revealing a faint planet signal. Future space-based telescope missions, such as the Habitable Worlds Observatory, (see also the Habitable Exoplanet Observatory (HabEx) and the Large UV/Optical/IR Surveyor (LUVOIR) concepts[1, 2]) will focus on the detection and characterization of exo-Earths around Sun-like stars. Such missions will require raw contrasts of $\simeq 10^{-10}$ at a close angular separation of $\approx 0.1"$, which induce strict wavefront quality and stability requirements[3]. The next generation of ground-based extremely large telescopes (ELTs) will target exo-Earths around low-mass stars. While the raw contrast requirements are less stringent ($\simeq 10^{-5}$ at $\approx 0.015"$), ELTs must also deal with turbulent residuals appearing downstream of the extreme adaptive optics system [4].

Assuming the planet position is well constrained, the coronagraph can be connected to a medium-to-high resolution spectrograph via an optical fiber to enhance rejection of starlight at small inner working angles[5, 6]. Furthermore, high dispersion coronagraphy (HDC)[5, 6] can be used to measure planetary orbital velocities and atmospheric compositions[7]. Modern ground-based instruments such as the Keck Planet Imager and Characterizer (KPIC) and the Subaru Telescope’s Rigorous Exoplanetary Atmosphere Characterization with High dispersion coronography (REACH) demonstrate the feasibility of HDC on diffraction-limited telescopes by coupling planet light into a single-mode fiber (SMF)[8, 9]. Additionally, next generation ground-based HDCs, such as the Planetary Systems Imager (PSI) on the Thirty Meter Telescope (TMT) are in development and future space-based HDC concepts are being considered[10, 11].

The SMF further rejects starlight due to its mode selectivity[12, 13]. However, despite the increased stellar suppression, the signal-to-noise ratio (SNR) of the planet spectrum remains limited due to residual starlight leaking into the fiber. This leakage results in excess photon noise and contaminates the planet signal[14, 15].

Wavefront control (WFC) is aimed at suppressing stellar speckles to help recover the companion’s signal. However, before WFC can be performed, the electric field must be measured in the focal plane. Two common methods for recovering the focal plane electric field are pair-wise probing (PWP), which utilizes phase diversity and the self coherent camera (SCC), where a pinhole is added in the Lyot Stop plane, creating fringes in the image plane that spatially encode the stellar speckles[16, 17, 18]. In WFC, one or two deformable mirrors (DMs) modify the incoming electric field to create a dark hole (DH) region, free of speckles[12]. While many wavefront control techniques exist, Electric Field Conjugation (EFC) is often used due to its superior performance relative to most other controllers[19]. EFC is the primary controller for the Nancy Grace Roman Space Telescope [20]. Additionally, EFC has been implemented on-sky with the VLT/SPHERE instrument[21]. EFC involves minimizing the focal plane electric field, using an optimal DM shape. The DM pattern is derived from a detailed instrument model[16].

The model-dependency of EFC limits its feasibility on-sky, as any telescope system instabilities will reduce the algorithm’s effectiveness [22, 23, 21]. Moreover, space-based telescopes too suffer from model uncertainties due to optical misalignments on-orbit caused by changing thermal gradients[24]. Implicit EFC (iEFC) utilizes a linear response between the DM and modulated intensity measurements, such that the reconstructed electric field is no longer computed explicitly[22, 25, 26]. iEFC is not the only measurement-based WFC technique. On the In Air Contrast Testbed (IACT) at NASA JPL, SCC was combined with EFC to create another empirical WFC method. A comparison between iEFC, SCC+EFC, and PWP+EFC showed similar performance ($\simeq 10^{-8}$) with a Vector Vortex Coronagraph (VVC)[27]. In this work, we propose an algorithm based on iEFC to minimize the speckles coupling into a SMF for both ground- and space-based systems. In Section 2, we derive the equations for fiber-based iEFC. In Section 3, we describe our simulation results, and in Section 4, we discuss and conclude our work.

To assess performance of iEFC through a SMF, we developed an end-to-end optical simulation based on the current layout of Caltech’s High Contrast Spectroscopy Testbed (HCST)[29].

Our simulation was constructed with the Fast Linearized Coronagraph Optimizer (FALCO) package within MATLAB, which provides realistic optical propagations for coronagraphs[30]. All simulation trials were computed with a central wavelength of $\lambda_{0}=750~{}nm$.We simulated a 55 segment off-axis telescope pupil, based on the LUVOIR-B design, with a circularly clipped outer edge. The circumscribed telescope diameter is 7.989 meters with each segment measuring 0.955m flat-to-flat. The segment gaps are $<0.1\%$ of the pupil diameter[31]. We also simulated a Lyot stop with a diameter of 82% of the incoming beam diameter, a charge 6 VVC, a 34x34 actuator Boston Micromachines DM, and a single-mode fiber with a mode field radius of $0.507~{}\lambda_{0}/D$.

In both EFC and iEFC simulations, we used 2 DM probes. The first DM probe is a sine wave, generated by summing over all of the even Fourier modes in our defined control region. The second DM probe is a cosine wave, generated by summing over all of the odd Fourier modes within our control region. By creating both sine and cosine probes, we capture both the real and imaginary components of the electric field.

Realistic speckles were generated using a phase map spanning low-high spatial frequencies (nolls 4-100) with an RMS wavefront error (WFE) of 10 nm–consistent with the aberrations we observe on our testbed[32]. We neglect the impact of segment phasing errors for our study, instead focusing on WFE that spans the pupil as a whole.

In defining contrast performance, we adopt the formalism of Ref. 12. We define SMF normalized intensity (NI) as the measured power at the output of the SMF divided by measured intensity at the output of the SMF centered on the non-coronagraphic stellar PSF.

We first simulate fiber-based iEFC and EFC at 2 different locations within our image, using a circular control region of 1 $\lambda_{0}/D$ to match the dimensions of our fiber and a fixed configuration of 24 Fourier modes. This mode configuration was chosen so as to minimize edge effects. Four Fourier modes fall inside the nulled region where the fiber is centered (Fig. 4), however, we choose to oversize this area to 12 Fourier mode pairs, or 24 modes, in order to control spatial frequencies at the border of the dark hole.

A simulated stellar PSF is shown in Fig. 2. Note that fiber-based iEFC will not necessarily create a DH at a given fiber position, as it is minimizing the overlap integral rather than the electric field [12]. In Fig. 3, we compare performance of iEFC to that of EFC in polychromatic light with a $\Delta\lambda/\lambda_{0}=20\%$ bandwidth, 9 sub-bandpasses, and a circular control region with a diameter of 1 $\lambda_{0}/D$.

All solutions converge on a NI of $\approx 10^{-10}$ or lower in fewer than 8 iterations. Additionally, we observe similar performance between EFC and iEFC in simulations. The similarities in achieved contrast between EFC and iEFC are consistent with Ref 22’s conclusions. We also observe some variation in the NI as a function of the fiber position. This is to be expected, as the phase pattern of the electric field along the fiber tip, the orientation of the Fourier modes with respect to the fiber, and the WFE all result in variation of the NI.

The segmented LUVOIR-B pupil used in our simulations may produce non-linear cross talk between Fourier modes. Hence, more modes would be necessary if certain modes contribute more power to the WFC solution than others. To examine the impacts of non-linear cross talk, we simulate three different Fourier mode configurations at a fixed fiber location of (0, -8) $\lambda_{0}/D$. Fig 4 shows the Fourier modes overlaid on a 2-D projection of the fiber’s fundamental mode. The modes are spaced at intervals of 1 cycle/pupil. In Fig. 5, we compare the NI of EFC to that of iEFC for a varying number of Fourier modes and a bandwidth of $\Delta\lambda/\lambda_{0}=30\%$ over 13 sub-bandpasses. Over 20 iterations, EFC and iEFC again exhibit similar similar performance through the fiber, however, iEFC achieves higher contrasts (NI$\approx 10^{-11}$) as we increase the number of modes. iEFC with 112 modes is slower than that of the 64 mode configuration, yet both solutions converge on an NI of $\approx 10^{-11}$. The results of Fig. 5 demonstrate that past 64 modes, there is no benefit to increasing the number of modes, as the system edge effects become negligible. However, additional work is needed to identify the optimal mode configuration for the iEFC controller. With a more optimal mode selection, we may be able to reduce the number of modes needed for the Jacobian measurement, thus decreasing the measurement time.

While we find that iEFC contrast improves with the number of Fourier modes, it is important to consider the increased Jacobian measurement time for iEFC in this analysis. In Fig. 6, we estimate the time to construct the Jacobian in iEFC vs EFC for real observations. We assume an arbitrary $\Delta\lambda/\lambda_{0}=20\%$ bandwidth with 9 sub-bandpasses. We also assume that the detector readout time is negligible. For EFC, the Jacobian construction time is obtained directly from our FALCO simulation. In iEFC, we estimate the Jacobian construction time, assuming 1-second integrations.

We observe that as the number of modes are increased, the time to construct the iEFC measurement Jacobian scales significantly. On the other hand, the EFC Jacobian construction time remains fixed, as it is computed from a model with a fixed number of modes. The measurement time of the iEFC response matrix can be prohibitive for certain operations in which the time scale of the WF variations is similar to that of the response matrix. However, this should not pose a constraint when the coronagraph instrument is stable enough such that a single response matrix can reflect the status of the WFE over an extended period of time. Future space-based imaging missions, in particular, will require such stability as well as a robust wavefront sensing and control loop to reach at least $10^{-10}$ contrast[2]. Thus, the choice of mode configuration becomes a compromise between the desired contrast level and the Jacobian computation time. Note that, for EFC, the issue of long Jacobian computation times is not relevant, as WFE is not included within the optical model. Also note that Fig. 6 does not account for the additional time required for wavefront sensing (WFS) with EFC. The estimated EFC WFS time is $\approx 54$ seconds which is shorter than the respective Jacobian computation time.

When implementing EFC and iEFC on ground-based instruments, the model errors are much larger than those of space-based instruments due to the presence of a turbulent atmosphere. In Fig. 7, we examine how EFC and iEFC respond to model errors within our simulation over a $\Delta\lambda/\lambda_{0}=30\%$ bandwidth and 13 sub-bandpasses. These errors are roughly consistent with what we would observe on ground-based telescopes[33]. To generate model errors, we use the phase map shown in Fig. 1 and vary the amplitude of RMS WFE. For iEFC, we use a fixed configuration of 24 Fourier modes. Both algorithms converge on a solution within 9 iterations until $\approx 150$ nm RMS WFE at which point EFC begins to diverge. Furthermore, as the WFE amplitude is increased, the contrast achieved with EFC decreases. This is consistent with what we expect as EFC computes a control solution from an ideal optical model. However, for iEFC in the high WFE regime, the solution remains fairly constant, with iEFC showing a $\approx$100x contrast improvement over EFC with 150 nm RMS WFE.

We describe a new method of focal plane wavefront control through a SMF that is applicable to both ground- and space-based telescopes. The model-free approach in iEFC combined with the mode selectivity of a SMF provides a simple method of wavefront control at higher contrasts than that of conventional EFC or iEFC[22, 12].

In simulations, we consistently demonstrate broadband contrasts of at least $10^{-10}$ with a charge 6 VVC in the low model error regime and contrasts of $\approx 10^{-9}$ in the high model error regime. We observe similar performance between EFC and iEFC through the fiber when model errors are small, but the performance of EFC degrades when simulating RMS WFE that is consistent with what we would see on a ground-based telescope. This is a particularly notable result, as EFC performance on ground-based telescopes may be limited by the model-based Jacobian solution[23]. Thus, high contrasts with fiber-based iEFC may be achievable on current and future ground-based fiber-fed spectrographs such as the Keck-KPIC, Keck-HISPEC, and TMT-MODHIS instruments [8, 34]. iEFC is also a promising control algorithm for space-based telescopes, as it eliminates the challenges associated with modelling on-orbit optical degradations[24].

We observe that a greater number of modes correlates with a higher contrast in iEFC through a SMF up to 64 modes. Beyond this modal configuration, the edge effects become negligible, as our solutions converge on an NI of $\approx 10^{-11}$. The improved performance with 64 Fourier modes could result from a less-than ideal modal selection. The effectiveness of individual modes may be reduced, as there are no trivial modes that would be orthogonal for a segmented pupil. This principle can be extended to EFC as well. Selecting the orthogonal modes could also make it more difficult for EFC and iEFC to converge. Through our runtime comparisons of the Jacobian between EFC and iEFC, we observe that the choice of mode configuration is a trade-off between the desired contrast level and the Jacobian construction time. This result is particularly notable for future space-based imaging systems, as the coronagraph instrument will need to remain well-aligned over the duration of the WFC calibrations.

Future work will be needed to identify the optimal number of modes in our iEFC control implementation. By optimizing the modal selection within iEFC, we may be able to reduce the number of modes needed to construct the measurement Jacobian, thus decreasing the time required for calibrations. We will also investigate how changing the Fourier mode spacing might affect our WFC solution. Additionally, we plan on developing a more robust simulation for iEFC on ground-based telescopes. If the interaction matrix is measured on-sky, we must account for effects such as temporal wavefront errors occurring over our calibration timescale. Such errors may impact our final WFC solution. Furthermore, we plan on conducting fiber-based iEFC experiments using the fiber injection unit on HCST. Due to its model-free approach and ease of implementation, iEFC through a SMF is a promising path forward for achieving high contrasts in future ground and space-based missions.