# Wavefront Sensing by a Common-Path Interferometer for Wavefront Correction in Phase and Amplitude by a Liquid Crystal Spatial Light Modulator Aiming the Exoplanet Direct Imaging

^{*}

## Abstract

**:**

## 1. Introduction

^{3}–λ/10

^{4}.

^{−9}m from fabrication defects and misalignments introduce a bright starlight speckle halo around the theoretical stellar image that overwhelms the faint image of an orbiting planet or a circumstellar disk. This speckle halo evolves in response to minute changes in the thermal and mechanical state of the observatory. Stellar coronagraphs rely critically on closed-loop wavefront sensing and have to compensate for these aberrations in time. Therefore, space-based telescopes require on-orbit options for wavefront sensing and its control, where all computations associated with the sensing and control algorithms are carried out by the flight computer aiming to achieve the deepest possible coronagraphic contrast.

^{3}(to 10

^{6}); this represents a substantial computational capacity. Here, fast linear least-squares coronagraph optimization (FALCO) [16] utilizes a coronagraph model highly optimized for speed showing practical improvement. Aiming at optimization, there are several more efforts known, such as reverse-mode algorithmic differentiation (RMAD) [17], pair-wise probing [18], Kalman filtering [19], and the self-coherent camera [20]. A coronagraph inverse problem can be solved to enable multiwavelength wavefront control by adding intensity constraints for each control wavelength of interest [21].

## 2. Materials and Methods: Common-Path Rotational-Shear Interfero-Coronagraph for Wavefront Measurement

#### 2.1. Common-Path Rotational-Shear Interfero-Coronagraph Principles

^{−8}= (Stellar PSF peak)/(Planet PSF peak) are visually saturated (or overwhelmed) by starlight leaked at the rotational shears of ψ = 180°, 90°, and 45°. A planet at a 5∙λ/D separation with a ε = 10

^{−8}contrast can be clearly visualized at the rotational shear less than ψ = 20°; see corresponding panels. By the smaller angle of rotation shear (e.g., 5°), the planet images also become suppressed.

_{2}(a) or after polarizing beam splitter PBS

_{2}in (b,c), we constructed a closed loop for a common optical path via successive mirror reflections. Indeed, two interferometer arms share the same path, are contrary directed, and, therefore, they have no optical path differences. Such a zero-difference in optical path lengths allows the achromatic functioning of the coronagraph. Rotational shear architecture achieves maximum transmission in the absence of pupil shielding. Shown in Figure 2, designs were tested under laboratory conditions, and their functionalities as achromatic coronagraphs were lab demonstrated [11,15].

#### 2.2. Wavefront Measurement by Means Interfero-Coronagraph and Phase Shifting Interferometry

**α**,

**β**) on a celestial sphere. Setting Cartesian coordinates, denoted as $\left(\mathit{x},\mathit{y}\right)$, in the image plane conjugated to the focal (or CCD) plane can be found by $\left(x=\mathrm{sin}\alpha \mathrm{cos}\beta ,y=\mathrm{sin}\alpha \mathrm{sin}\beta \right)$. Complex amplitudes in the pupil $\left(\mathit{u},\mathit{v}\right)$ and in the image $\left(\mathit{x},\mathit{y}\right)$ are Fourier transformed: $\widehat{P}\left(x,y\right){\mathrm{e}}^{i\widehat{\mathsf{\varphi}}\left(x,y\right)}={F}^{-1}\left\{P\left(u,v\right){\mathrm{e}}^{i\mathsf{\varphi}\left(u,v\right)}\right\}$.

## 3. Results

#### 3.1. Numerical Simulation of Wavefront Control in an Optical Scheme with an Interfero-Coronagraph

_{1}) and optical elements (lenses L

_{2..6}) in order to make several optical planes (with co-ordinates), Σ′ = (u′, v′), Σ″ = (u″, v″), and Σ‴ = (u″, v″) being optically conjugated to the primary pupil Σ

_{0}= (u, v). Aberrations of the telescope are resumed in Σ

_{0}. AO element—LC SLM is installed in the secondary pupil Σ′. Interfero-coronagraph is placed in the space before the tertiary pupil Σ″. In the next pupil Σ″″, we place CCD

_{2}, which is used to measure wavefront by recording a series of PSI images ${I}_{1..L}^{}$. Another CCD

_{1}is the field camera to observe an image in the focal plane F″, which is illuminated (shown in blue) by some beam splitters, assumed to be dichroic or switchable.

_{0}, Σ′, … Σ‴. Optical surface defects can be optically characterized with power spectral density (PSD) for micro-roughness; these surface figures produce Fresnel-propagation-like aberrations which we can associate with non-common-path aberrations (NCPA). Strictly, the NCPAs have the difference from the aberrations detected by a wavefront sensor (WFS), important in that NCPAs degrade coronagraphic image quality. In our case, because wavefront measurement is organized after the coronagraph, NCPAs (Fresnel aberration screens) generate not only phase errors but also amplitude wavefront errors. In the model, NCPA can be introduced as a phase aberration screen in an intermediate Σ

_{NCPA}plane, which is a non-conjugate optically to the pupil. By Fresnel propagation, NCPA results in a high spatial frequency amplitude (and phase) modulation of the pupil. Therefore, NCPAs’ effect can be studied if one considers several aberration screens in mutually non-conjugated planes (Σ

_{NCPA1}, ...). In Figure 5, we show a single Σ

_{NCPA}plane.

_{0}and Σ

_{NCPA}are shown in Figure 6a and Figure 6b, respectively. These aberrations do not include any classic geometric aberrations (coma, spherical, astigmatic, etc.) as represented by low-order Zernike polynomial decomposition. The shown aberrations consist only of micro-roughness errors, and they visually look like sky clouds. PSD was characterized by the local phase error $\mathsf{\delta}\mathsf{\varphi}~{\mathsf{\rho}}^{-2}$, where $\mathsf{\rho}=\sqrt{{u}^{2}+{v}^{2}}$ is the radial scale. The RMS error σ = 10 nm of the wavefront ${\mathsf{\varphi}}_{{\Sigma}_{0}}\left(u,v\right)$ in Σ

_{0}, and the RMS σ = 1 nm error of the ${\mathsf{\varphi}}_{{\Sigma}_{\mathrm{NCPA}}}\left(u,v\right)$ wavefront in Σ

_{NCPA}were calculated at the characteristic length of $\mathsf{\rho}=D/2$.

_{1}field camera. A companion (planet) was computed with the PSFs peak-to-peak contrast C = 10

^{−9}between the point-like light sources planet and star. Contrast C was modeled as $C=\frac{{I}_{*,@\lambda}}{{I}_{p,}@\lambda}$ and the wavelength λ = 500 nm. The star (*) and planet (p) were separated at a 5 λ/D stellocentric distance. Panels (c), (d) were calculated with the wavefront distortions ${\mathsf{\varphi}}_{{\Sigma}_{0}}\left(u,v\right)$ by σ = 10 nm. In panel (c), the coronagraphic image I(x″, y″) was computed without any wavefront correction. The image contains strong speckle effects, which completely masks the faint planet image. In panel (d), the coronagraphic image was computed as phase corrected, and we applied the phase screen of $-{\mathsf{\varphi}}_{{\Sigma}_{0}}\left({u}^{\prime},{v}^{\prime}\right)$ by AO LC SLM in the plane Σ′(u′, v′). Now, both the planet PSF and its ghost symmetric copy were completely cleared of speckles. This is because the optically conjugated planes Σ

_{0}and Σ′ are considered to contain the equal phase distributions, which were compensated for. In panels (e) and (f), besides the initial aberration ${\mathsf{\varphi}}_{{\Sigma}_{0}}\left(u,v\right)$, an additional ${\mathsf{\varphi}}_{{\Sigma}_{\mathrm{NCPA}}}\left(u,v\right)$ aberration in different and not conjugated plane Σ

_{NCPA}was computed. In panel (e), correction was not applied, and this image is nearly similar to that in panel (c). In panel (f), correction was applied by setting the phase screen $-{\mathsf{\varphi}}_{\Sigma \u2019}\left({u}^{\prime},{v}^{\prime}\right)$ in the plane Σ′(u′, v′); note that $-\mathsf{\varphi}{\prime}_{{\Sigma}^{\prime}}\left({u}^{\prime},{v}^{\prime}\right)\ne -{\mathsf{\varphi}}_{{\Sigma}_{0}}\left({u}^{\prime},v\prime \right)$, which is non-equal to the correction used for result in (d) panel. Moreover, the amplitude distribution ${P}_{{\Sigma}^{\prime}}\left({u}^{\prime},{v}^{\prime}\right)$ (not shown) becomes non-uniform across the pupil.

_{NCPA}. Shown in panel (g), the radially averaged profile of coronagraphic image (f) robustly visualizes a companion in 5 λ/D by the intensity maximum.

#### 3.2. Lab Experiment to Verify Wavefront Correction

#### 3.2.1. Schematics of Lab Experiment

_{2}, a linear polarizer, and through a reflecting phase-only LC SLM propagating in the direction of the AIC-180°interfero-coronagraph. Lenses L

_{3}–L

_{4}conjugate optically the pupil plane Σ′ where the LC SLM is installed to the beam combiner plane of the AIC-180° interfero-coronagraph, where interfering waves are superposed. After the coronagraph, another beam splitter BS is located. In this direction after the beam splitter reflects, an image (focal) plane is formed and is observed by a CCD

_{1}field camera. CCD

_{1}stands to observe a coronagraphic image in the focal plane. In the other direction, the beam splitter transmits an image, and an additional lens L

_{6}stands to form an optical pupil plane image. This pupil image is observed by the second CCD

_{2}to measure the wavefront distribution according to the PSI algorithms; see Table 1.

#### 3.2.2. Wavefront Correction Maps in Pupil Plane

_{2}applying PSI algorithms (modified as shown in Figure 3) are shown in Figure 8.

_{NCPA}).

#### 3.2.3. Coronagraphic PSF Suppression in Focal Plane

_{1}, where coronographic images were recorded. The non-coronagraphic and coronagraphic images shown in Figure 9a,b correspondingly were compared to measure a coronagraphic contrast. To obtain a non-coronographic image (after the coronagraph), the input wavefront contains a step-form phase modulation, and the phase gap between the left and right half-planes was about π (see Figure 4: ${\mathsf{\phi}}_{1}-{\mathsf{\phi}}_{2}=\mathsf{\pi}$ ). By this method, the non-coronagraphic image looks similar to a theoretical PSF as the diffraction image of a point-like source with clearly observed airy rings. The non-coronagraphic frame was captured with 0.02 ms exposure.

_{1}matrix (see CCD

_{1}in the optical scheme in Figure 7) by 30 ms exposure of the coronographic image. Therefore, the represented non-coronagraphic(Figure 9a) and coronagraphic images (Figure 9b) are shown with their scale respective of their exposure ratio 30/0.02 = 1500.

^{5}above the PSF value at the maximum, (see black dashed line). Post-processing can improve this value by an order of magnitude, which has not yet been shown in the presented laboratory experiment and will require a greater number of processed images.

#### 3.3. Constraints and Their Overcome

#### 3.3.1. Using an LC SLM for Extreme Wavefront Correction

^{6}compared to the lower number of DM actuators ~10

^{3}...10

^{4}. (ii) An amplitude-phase modulation and a general polarization modulation are both possible when using polarization devices. Amplitude-phase modulation occurs, e.g., when two linear polarizers are placed, one before and one after the LC SLM, and polarizers are oriented at their transmission axis at different angles from the direction of the main working polarization axis of the LC SLM. (iii) It is possible to realize an arbitrary wavefront surface, including wavefront discontinuities, which is practically impossible with the DM.

^{8}(i.e., λ/256), at which the corrected wavefront does not have a sufficient quality for observing exoplanets. However, AO LC SLM phase modulation accuracy can be enhanced if an initial phase distortion is within, e.g., λ/10 P-V and has to be corrected. Such a calibrated AO LC SLM performs λ/2500 ≈ 0.1 π/2

^{8}radian accuracy.

#### 3.3.2. Using an LC SLM for Wavefront Correction in Phase–Amplitude Mode

_{1}and φ

_{2}, it becomes possible to compensate for the initial phase and amplitude modulation in these pixels ${A}_{01}{e}^{i{\mathsf{\phi}}_{01}},{A}_{02}{e}^{i{\mathsf{\phi}}_{02}}$. By φ

_{1}and φ

_{2}control, we can equalize the optical fields in pixels in the form of the equal complex amplitudes and phases. When solving the following equation:

_{1}and φ

_{2}, reducing the number of variables from four to two: $\u2206\mathsf{\phi}={\mathsf{\phi}}_{01}-{\mathsf{\phi}}_{02}$ means the difference of initial phases and $a={A}_{01}/{A}_{02}$ means the ratio of amplitudes:

## 4. Discussion

^{9}, requires a correction accuracy better than λ/500 on controllable 500 × 500 pixels.

^{5}at a stellocentric distance of more than 2 diffraction radii (~2 λ/D) at a wavelength of 633 nm.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Rotation-shear interfero-coronagraph: principles and simulated output. (

**a**)—anti-phase superposition on-axis for stellar PSF (asterisk denoted) and doubling off-axis planet’s PSF (dot denoted). (

**b**)—monochromatic starlight incomplete suppression, leakage considering the stellar size of Θ = 0.01 λ/D (@ D = 2.4 m, λ = 500 nm) by various rotational shears (the angles of rotation are noted), the shown planet PSF considered at the stellocentric separation of ρ

_{0}= 5 λ/D, and contrast ε = 10

^{−8}. (

**c**)—in polychromatic band (λ = 350–850 nm), picture of Solar System as observed from 10 pc with the telescope D = 2.4 m and the interfero-coronagraph having 3.6° angular shear; Jupiter, Saturn, Uranus, Neptune seen visualized toward periphery.

**Figure 2.**Optical schematics of common-path nulling interferometers with various rotational-shear angles acting as interfero-coronagraphs (on their dark port): (

**a**) CP-AIC (RSI-180°) with rotational-shear angle fixed at 180°; (

**b**) RSI-VAR with a variable rotational-shear angle; (

**c**) RSI-10° rotational-shear angle fixed at 10°.

**Figure 3.**Phase modulation provided in the pupil semi-plane. Denoted are intensity values ${I}_{\mathrm{1..3}}^{\left(\mathrm{u}\right)},{I}_{\mathrm{1..3}}^{\left(\mathrm{d}\right)}$ used by the three-step PSI; the subscripts (u) and (d) stand for the upper and lower halves of the pupil; an additional phase modulation with the corresponding intensities of ${I}_{4}^{\left(\mathrm{u}\right)},{I}_{4}^{\left(\mathrm{d}\right)}$ for the four-step PSI algorithm is in blue.

**Figure 4.**Illustration (in left column) for the measurement of wavefront by 180° interfero-coronagraph and (in right column) for the correction of wavefront by an AO LC SLM placed before the 180° interfero-coronagraph. (Note that here the corrected “left” side of SLM is shown. The same result could be achieved with “right” side just by altering the sign before ${\mathsf{\phi}}_{1}-{\mathsf{\phi}}_{2}$).

**Figure 5.**Optical scheme to assemble telescope by AO, interfero-coronagraph, field camera, and pupil camera for PSI technique.

**Figure 6.**Simulated phase aberrations and corresponding coronagraphic images. Aberrations: (

**a**)—phase distribution ${\mathsf{\varphi}}_{{\Sigma}_{0}}\left(u,v\right)$ with σ = 10 nm rms in Σ

_{0}plane and (

**b**)—${\mathsf{\varphi}}_{{\Sigma}_{\mathrm{NCPA}}}\left(u,v\right)$ with σ =1 nm rms in Σ

_{NCPA}. (

**c**–

**f**)—evaluated coronagraphic images I(x″, y″) at focus F″ (see optical scheme in Figure 5) on CCD

_{1}field camera, computed with the PSFs peak-to-peak contrast between point-like light sources: the planet and the star C = 10

^{−9}. (

**c**,

**d**)—calculated with the wavefront distortion of ${\mathsf{\varphi}}_{{\Sigma}_{0}}\left(u,v\right)$. (

**c**)—non-corrected wavefront. (

**d**)—corrected by setting the phase screen $-{\mathsf{\varphi}}_{{\Sigma}^{\prime}}\left({u}^{\prime},{v}^{\prime}\right)=-{\mathsf{\varphi}}_{{\Sigma}_{0}}\left(u,v\right)$ in the plane Σ′(u′, v′); here, the planet PSF and its symmetric copy were completely cleared from the speckles. In panels (

**e**,

**f**), both aberrations ${\mathsf{\varphi}}_{{\Sigma}_{0}}\left(u,v\right)$ and ${\mathsf{\varphi}}_{{\Sigma}_{\mathrm{NCPA}}}\left(u,v\right)$ in the different planes Σ

_{0}and Σ

_{NCPA}were computed. (

**e**)—any wavefront correction was not applied. (

**f**)—wavefront correction was applied by setting phase screen $\u2013\mathsf{\varphi}{\prime}_{{\Sigma}^{\prime}}\left({u}^{\prime},{v}^{\prime}\right)\ne -{\mathsf{\varphi}}_{{\Sigma}_{0}}\left(u,v\right)$ in the plane Σ′(u′, v′). (

**g**)—radially averaged profile of coronagraphic image of panel (

**f**).

**Figure 7.**Optical scheme (

**a**) and photo (

**b**) of the lab experiment for wavefront correction with measurement of wavefront distortions after an interfero-coronagraph. Marked: laser, spatial filter: L

_{1}lens, L

_{2}collimating lens, linear polarizer, phase-only LC SLM (LC SLM shown in the in the transmitting mode in panel (

**a**)), interfero-coronagraph (AIC-180°) (shown in enlarged size in tab (

**b**)), beam splitter, focusing lenses L

_{3}–L

_{5}, CCD

_{1}(field camera in focal plane), focusing lens L

_{6}to form a pupil plane, CCD

_{2}(camera in pupil plane).

**Figure 8.**Correction of wavefront in left (right) semi-plane in pupil: (

**a**)—uncorrected wavefront measured by interfero-coronagraph. (

**b**,

**c**)—consequently corrected wavefront in left and right semi-planes (the area within the shown red circle was used as corrected pupil to form focal images in following Figure 9). (

**d**)—wavefront phase error power spectrum density (PSD) averaged over radial cross-section; before correction—blue (dashed) line, after correction—red (solid) line.

**Figure 9.**Experimental images: (

**a**)—non-coronagraphic image (with exposure 0.02 ms). (

**b**)—coronagraphic image by the phase correction down to with σ ≈ λ/40 (with exposure ≈ 30 ms). (

**c**)—averaged radial cross-sections: blue (solid) line for the non-coronagraphic image (

**a**), red (solid) line for the coronagraphic image (

**b**), blue (dashed) line for the radial cross-section of the theoretical PSF (non-coronagraphic), black (solid) line—cross section of another in time realization of the coronagraphic image, black (dashed) line—averaged cross section of the difference between two coronagraphic images.

PSI Technique | $\mathbf{Phase}\text{}\mathbf{Distribution}\text{}\mathbf{\varphi}\left(\mathit{u},\mathit{v}\right)$ | $\mathbf{Visibility}\text{}\mathbf{Distribution}\text{}\mathit{\gamma}\left(\mathit{u},\mathit{v}\right)$ |
---|---|---|

Three-step: $\u2206\mathsf{\varphi}$ = −π/2, 0, π/2 | $ta{n}^{-1}\left(\frac{{I}_{1}-{I}_{3}}{2{I}_{2}-{I}_{1}-{I}_{3}}\right)$ | $\frac{\sqrt{\left({\left({I}_{1}-{I}_{3}\right)}^{2}+{\left(2{I}_{2}-{I}_{1}-{I}_{3}\right)}^{2}\right)}}{{I}_{1}+{I}_{3}}$ |

Four-step: $\u2206\mathsf{\varphi}$ = −π/2, 0, π/2, π | $ta{n}^{-1}\left(\frac{{I}_{4}-{I}_{2}}{{I}_{1}-{I}_{3}}\right)$ | $2\frac{\sqrt{\left({\left({I}_{4}-{I}_{2}\right)}^{2}+{\left({I}_{1}-{I}_{3}\right)}^{2}\right)}}{{I}_{1}+{I}_{2}+{I}_{3}+{I}_{4}}$ |

N-step: $\u2206{\mathsf{\varphi}}_{i}$ = 0...2π/(N−1) | $ta{n}^{-1}\left(\frac{{{\displaystyle \sum}}_{i=1}^{N}{I}_{i}\mathrm{sin}\u2206{\mathsf{\varphi}}_{i}}{{{\displaystyle \sum}}_{i=1}^{N}{I}_{i}\mathrm{cos}\u2206{\mathsf{\varphi}}_{i}}\right)$ | $2\frac{\sqrt{{{\displaystyle \sum}}_{i=1}^{N}\left[{I}_{i}^{2}{\left(\mathrm{sin}\u2206{\mathsf{\varphi}}_{i}\right)}^{2}+{I}_{i}^{2}{\left(\mathrm{sin}\u2206{\mathsf{\varphi}}_{i}\right)}^{2}\right]}}{{{\displaystyle \sum}}_{i=1}^{N}{I}_{i}}$ |

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## Share and Cite

**MDPI and ACS Style**

Yudaev, A.; Kiselev, A.; Shashkova, I.; Tavrov, A.; Lipatov, A.; Korablev, O.
Wavefront Sensing by a Common-Path Interferometer for Wavefront Correction in Phase and Amplitude by a Liquid Crystal Spatial Light Modulator Aiming the Exoplanet Direct Imaging. *Photonics* **2023**, *10*, 320.
https://doi.org/10.3390/photonics10030320

**AMA Style**

Yudaev A, Kiselev A, Shashkova I, Tavrov A, Lipatov A, Korablev O.
Wavefront Sensing by a Common-Path Interferometer for Wavefront Correction in Phase and Amplitude by a Liquid Crystal Spatial Light Modulator Aiming the Exoplanet Direct Imaging. *Photonics*. 2023; 10(3):320.
https://doi.org/10.3390/photonics10030320

**Chicago/Turabian Style**

Yudaev, Andrey, Alexander Kiselev, Inna Shashkova, Alexander Tavrov, Alexander Lipatov, and Oleg Korablev.
2023. "Wavefront Sensing by a Common-Path Interferometer for Wavefront Correction in Phase and Amplitude by a Liquid Crystal Spatial Light Modulator Aiming the Exoplanet Direct Imaging" *Photonics* 10, no. 3: 320.
https://doi.org/10.3390/photonics10030320