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# Vortex coronagraph: revisiting the phase retrieval properties via Zernike
analysis

G. Orban de Xivry  Space sciences, Technologies, and Astrophysics Research
(STAR) Institute, Université de Liège, allée du Six Août 19c, 4000 Liège,
Belgium  O. Absil  Space sciences, Technologies, and Astrophysics Research
(STAR) Institute, Université de Liège, allée du Six Août 19c, 4000 Liège,
Belgium

###### Abstract

High contrast imaging (HCI) is fundamentally limited by wavefront aberrations,
and the ability to perform wavefront sensing from focal plane images is key to
reach the full potential of ground and space-based instruments. Vortex focal
plane mask coupled with downstream pupil (Lyot) stop stands as one of the best
small-angle coronagraphs, but is also sensitive to low-order aberrations.
Here, we revisit the behavior of the vortex phase mask, from entrance pupil
down to the final detector plane, with Zernike polynomials as input phase
aberrations. In particular we develop a second-order expansion that allows us
to analyze the phase retrieval properties in a more intuitive and accurate way
than previously proposed. With this formalism, we show how the azimuthal
vortex modulation modifies the phase retrieval properties compared to normal
imaging. In particular, our results suggest that images obtained with a scalar
vortex coronagraph can be used for unambiguous focal-plane wavefront sensing
in any practical situation. We compare our results with numerical simulations
and discuss practical implementation in coronagraphic instruments.

###### keywords:

High contrast imaging, coronagraphy, vortex phase mask, Zernike analysis,
phase retrieval

##  1 INTRODUCTION

The direct detection of exoplanets relies on coronagraphs to reach the highest
contrast. All of them are however limited by the imperfections of the entrance
wavefront. Among the various coronagraph designs, the vortex coronagraph
stands out as one of the best small-angle coronagraphs, making it also
particularly sensitive to low-order aberrations. Therefore, enabling wavefront
sensing directly at the focal plane is key to unlock the full potential of
future ground and space-based instruments.

A vortex coronagraph is composed of a vortex focal plane mask coupled with a
downstream pupil (Lyot stop). A schematic layout, Fig. 1, illustrates the text
book effect of the vortex coronagraph: to move the light of an on-axis source
outside the downstream geometric pupil. The diffracted light is then blocked
by the Lyot stop, providing theoretically a perfect starlight cancellation for
a circular entrance aperture. The vortex phase mask itself introduces a phase
screw $e^{il_{p}\phi}$ with $\phi$ the azimuthal coordinate, and $l_{p}$ the
topological charge of the vortex. There are two flavors of vortex
coronagraphs: vectorial (VVC) and scalar (SVC). A vector vortex is based on a
half-wave plate with spatially varying fast axis, which results in conjugated
phase ramps applied to the two circular polarization states. This is described
by the conjugated sign of the topological charge, $\pm|l_{p}|$. On the other
hand, a scalar vortex is based on longitudinal phase delay and is described by
a single sign of the topological charge. Alternatively, a scalar vortex can be
obtained by isolating one circular polarization of a vector vortex.

In order to perform focal-plane wavefront sensing, classical imaging systems
typically require a known even aberration diversity to resolve the ambiguity
between focal plane images and phase aberrations in the pupil plane. Here, we
study the effect of the vortex azimuthal modulation on the ambiguity between
focal plane images and phase aberrations. Specifically, we develop a second-
order analytical approximation to describe the electrical field down to the
focal plane. This formalism allows us to analyze the phase retrieval
properties in an accurate way, and to compare it to classical imaging and
numerical simulations. In Sec. 2, we summarize our analytical approach and
provide a few illustrations. In Sec. 3, we use our formalism to analyze the
phase retrieval properties of VVC and SVC, first by evaluating the ambiguity
on individual Zernike modes and then for the combination of two modes. In Sec.
4, we conclude our contribution.

Figure 1: Schematic design of a vortex coronagraph. From left to right:
Entrance circular pupil (red), intermediate focal plane with the vortex mask
(green), downstream Lyot pupil plane and stop (red), final image plane
(green).

##  2 SECOND-ORDER ZERNIKE EXPANSION FOR THE VORTEX CORONAGRAPH

The theoretical expansion we propose is based on three steps. First, we
perform a second order expansion of the entrance pupil wavefront in order to
express the electrical field as a complex linear combination of Zernike
coefficients. Second, we propagate the field from the entrance pupil plane to
the downstream Lyot plane, assuming a (perfect) vortex phase mask in the
intermediate focal plane. Third, we propagate the field down to the final
image plane, where the detector is placed. The entire formalism is implemented
in Symbolic Python (Sympy).

Step 1: Pupil plane second order expansion.

We perform a second order approximation of the entrance pupil wavefront

$\displaystyle\begin{split}E_{\textrm{pup}}&=\exp(i\Phi)\\\ &\approx
1+i\Phi-\dfrac{1}{2}\Phi^{2}\\\ &\approx
1+i\sum_{j=2}a_{j}Z_{j}-\dfrac{1}{2}\sum_{k=1}b_{k}Z_{k}\end{split}$ (1)

where the wavefront is expressed as a combination of Zernike polynomials. The
square wavefront $\Phi^{2}$ is also expressed as a (different) combination of
Zernike’s obtained by expressing the products of two Zernike polynomials as a
sum of Zernike’s. We use here the result of Marthar[1].

Step 2: Propagation to the Lyot plane.

After the pupil expansion, we perform the propagation from the pupil to the
Lyot plane, assuming a circular aperture and a perfect vortex phase mask of
even topological charge $l_{p}$ in the intermediate focal plane. The
linearized expression becomes

$E_{\rm Lyot}=i\sum_{j=2}a_{j}\zeta_{j}-\dfrac{1}{2}\sum_{k=2}b_{k}\zeta_{k}$
(2)

with
$\zeta_{j}=\mathcal{F}^{-1}\left[\mathcal{F}\left[Z_{j}\right]e^{jl_{p}\phi}\right]$
denoting the field distribution in the Lyot plane when the input field is
defined by the Zernike polynomial $Z_{j}$. To convert Zernike polynomial
$Z_{j}$ to $\zeta_{j}$, we use the analytical formalism developed by Huby et
al.[2]. Note that we have assumed a Lyot stop is used, which blocks entirely
the diffracted light resulting from the perfect plane component (piston mode).
Hence there is no first term, $\zeta_{1}$, in the equation.

Step 3: Propagation to the image plane.

The electrical field in the image plane is obtained by taking the analytical
Fourier transform of each Zernike polynomial, see for example Niu and Tian[3].
The final intensity is obtained by taking the square modulus of this linear
combination and by evaluating it on a specific grid $(r,\phi)$.

###  2.1 Analytical example: astigmatism

To illustrate the analytical formalism, its differences with the first-order
approximation, and the quickly increasing complexity, we calculate here the
analytical result for astigmatism alone with an amplitude $a_{5}$. In the
following, we assume a topological charge $l_{p}=2$.

The linear approximation provides the following results

$E_{\rm pup}\approx 1+ia_{5}Z_{5}\longrightarrow E_{\rm
lyot}=-a_{5}\frac{\sqrt{2}Z_{4}}{2}\longrightarrow E_{\rm
img}=a_{5}\frac{\sqrt{6}J_{3}\left(2\pi k\right)}{4\pi k}$ (3)

The resulting electrical field in the image plane is a pure radial function
composed of a single Bessel function. Consequently, this implies that both
positive and negative signs of the astigmatism would produce two identical
intensity images.

The second order approximation provides the following results

$\displaystyle E_{\textrm{pup}}$
$\displaystyle=(1-\frac{a_{5}^{2}}{2})+ia_{5}Z_{5}+a_{5}^{2}\left(-0.25\sqrt{3}Z_{4}-0.05\sqrt{5}Z_{11}+0.15\sqrt{10}Z_{14}\right)$
(4) $\displaystyle E_{\textrm{lyot}}$
$\displaystyle=-a_{5}\frac{\sqrt{2}Z_{4}}{2}$
$\displaystyle\mathrel{\phantom{=}}+a_{5}^{2}\left\\{-0.05\sqrt{5}\left(\frac{\sqrt{2}Z_{12}}{2}+\frac{\sqrt{2}iZ_{13}}{2}\right)-0.25\sqrt{3}\left(\frac{\sqrt{2}iZ_{5}}{2}+\frac{\sqrt{2}Z_{6}}{2}\right)\right.$
(5)
$\displaystyle\mathrel{\phantom{=}}\left.+0.15\sqrt{10}\left(0.5Z_{12}-\frac{iZ_{13}}{2}\right)\right\\}$
$\displaystyle E_{\textrm{img}}$
$\displaystyle=a_{5}\frac{\sqrt{6}J_{3}\left(2\pi k\right)}{4\pi k}$
$\displaystyle\mathrel{\phantom{=}}+a_{5}^{2}\left\\{-\frac{0.375i\sin{\left(2\phi\right)}J_{3}\left(2\pi
k\right)}{\pi k}+\frac{0.5i\sin{\left(2\phi\right)}J_{5}\left(2\pi
k\right)}{\pi k}\right.$ (6)
$\displaystyle\mathrel{\phantom{=}}\left.+\frac{0.375\cos{\left(2\phi\right)}J_{3}\left(2\pi
k\right)}{\pi k}+\frac{0.25\cos{\left(2\phi\right)}J_{5}\left(2\pi
k\right)}{\pi k}\right\\}$

The resulting field in the image plane is no longer purely radial, and the
images differ depending on whether the sign of astigmatism is positive or
negative.

###  2.2 Illustration: comparing first order and second-order approximation

We illustrate below the better representativeness of the second order
approximation. The difference between first and second order was already
observed for tip-tilt by Huby et al[2]. Here we generalize the second order
formalism to all Zernike’s and investigate the phase retrieval properties.

Figure 2: Comparison between first and ssecond order approximation for tilt,
focus, astigmatism ($Z_{3}$, $Z_{4}$, $Z_{5}$). In each case we use of scalar
vortex of topological charge $l_{p}=2$, and an aberration level of 1 rad rms.

The second order approximation reveals the asymmetry of the tilt ($Z_{3}$) and
the astigmatism ($Z_{5}$). An opposite sign for the astigmatism leads to a
rotation of 90 degrees of the pattern. Focus remains a purely radial function,
which can also be verified at the third order.

###  2.3 Comparison with numerical approaches

It is interesting to compare our second order approach with numerical
simulations to establish the validity range of our approximation, and
understand any numerical artifacts.

A numerical representation of a pure vortex screw on a finite grid is
challenging, especially close to the singularity (the center) where the phase
rapidly changes over a limited number of pixels. At least two approaches are
typically considered, individually or in combination, to obtain an accurate
representation of the phase ramp: oversampling the central region, and
numerically enforcing the perfect cancellation in the downstream pupil.

The first method involves computing the (central) phase ramp on a larger grid
and then rebinning the complex array down to the normal sampling of the
simulation. The method produces good results but not a perfect representation
of the vortex phase ramp. The second method allows to free the numerical
simulation from any aliasing and Fourier transform artifacts. Specifically, it
ensures that the perfect vortex with an unaberrated field is perfectly
simulated according to theory, i.e., the intensity inside the downstream
(Lyot) pupil plane is zero for a circular pupil. To enforce this, the perfect
field plus perfect coronagraph are propagated to the Lyot stop, then the
intensity is zeroed-out inside the pupil and back-propagated to the focal
plane. The resulting electrical field can then be used for future
propagations. In the following, we use a custom numerical simulator, which
only implements the second method (back-propagation), but not the
oversampling.

We perform a first visual comparison by injecting astigmatism with increasing
amplitudes up to 1 rad rms, see Fig. 3. The images are normalized by their
off-axis PSF, which allows a direct comparison between the two approaches and
informs us on the raw contrast. The morphology and the flux of the PSFs match
closely but begin to diverge at around 0.4 rad, reaching a factor of more than
2 in intensity at around 1 rad. This indicates that the second-order
analytical approach becomes invalid.

Figure 3: Vortex coronagraphic images comparing analytical (top) and numerical
simulations for different level of astigmatism between 1e-2 and 1 rad rms.

We now turn our attention to a pure radial mode: focus. We numerically
simulate the propagation of a small defocus through an SVC of charge 2 and
compare the images obtained for opposite signs of the mode. We perform the
same using the analytical approach. The differential images, normalized by the
peak intensity of one propagation, are illustrated in Fig. 4, with a
4$\lambda/D$ radial field-of-view and 32 pixels per $\lambda/D$. While the
difference for the analytical approach leads to random numerical noise only,
the Fourier-based simulations lead to a distinctive pattern which would imply
that there is no sign ambiguity on this mode. This is a numerical artifact due
to the finite sampling of the vortex phase ramp, and the entrance and Lyot
pupils. Unsurprisingly, we expect that the sampling precision, especially in
the central region of the vortex screw, will always limit the fidelity of
vortex coronagraphic simulations to some extent and require dedicated
attention.

Figure 4: Difference between images with opposite sign of focus. (Left) based
on numerical simulations of a charge-2 SVC. (Right) based on our second order
analytical approach.

##  3 SIGN AMBIGUITIES WITH VORTEX CORONAGRAPHS

###  3.1 Individual Zernike modes

For circularly symmetric pupils and classical imaging, there is no unique
relationship between focal plane images and phase aberrations in the pupil
plane. For point-like sources, the non-uniqueness consists in a (single) sign
ambiguity on even modes.

Based on this consideration, to evaluate the ambiguity between two images
(obtained with classical or coronagraphic imaging) with phase aberrations
$\phi_{1}$ and $\phi_{2}$, we use a simple L2 sensitivity metric based on the
final images. A null value means the images are identical. For non-zero
values, we can expect that the higher the better to perform phase retrieval.
We compute the sensitivity for the first 13 Zernike modes starting from tilt
($Z_{3}$), using an amplitude of 0.1 rad. We perform the computation both for
a scalar and a vector vortex of topological charge 2, as well as for non-
coronagraphic imaging as a reference. Similarly to normal imaging, we can see
that the vector vortex is insensitive to the sign of even modes. The scalar
vortex displays a different behavior, where only the pure radial modes are
ambiguous (here focus $Z_{4}$ and spherical aberration $Z_{11}$), and the
other even modes are unambiguous (here, oblique astigmatism $Z_{5}$ and
vertical astigmatism $Z_{6}$, but also $Z_{12}$, $Z_{13}$, $Z_{14}$, and
$Z_{15}$). For non-even modes, SVC and VVC display the same sensitivity with
some slight discrepancies from classical imaging.

Figure 5: Sensitivity to individual modes, from tilt ($Z_{3}$) to $Z_{15}$.
The grey shaded area denotes even Zernike modes. $Z_{4}$ (focus) and $Z_{11}$
(spherical) are pure radial modes.

This phase retrieval property ensues from the azimuthal modulation as opposed
for example to the classical focus phase diversity, which introduces a radial
modulation. It follows that pure radial modes (e.g., focus and spherical) are
not modulated by the phase screw in an informative way and the sign ambiguity
cannot be lifted.

###  3.2 Combination of Zernike modes

With the azimuthal modulation introduced by the vortex, the ambiguity on even
modes breaks down and we can anticipate that with a combination of modes the
ambiguity on pure radial modes may be lifted. We consider a combination of
focus and astigmatism with equal amplitudes ($0.5/\sqrt{2}$ rad rms each). By
considering all sign combinations of those two modes, we obtain 4 different
images, see Fig. 6. Two pairs are closely similar but not identical, as
observed in the zoom insets and illustrated by the image difference Fig. 7.
From these illustrations, we can consider the ambiguity lifted, albeit with
presumably lower sensitivity. Arguably, this conclusion can be extended to any
combination of modes, suggesting that sufficient information exists to lift
sign ambiguities in the focal plane image of a scalar vortex coronagraph.

Figure 6: (Left) Four images obtained with a combination of $Z_{4}$ and
$Z_{5}$ with equal rms of $0.5/\sqrt{2}$. The zoom insets emphasize the subtle
difference.  Figure 7: Subtraction of the two close pairs, $(I_{1}-I_{4})$ and
$(I_{2}–I_{3})$, illustrated in Fig. 6.

##  4 CONCLUSION

In conclusion, our results suggest that phase aberrations can be retrieved in
any practical scenario using scalar vortex coronagraph (SVC) images. This
finding corroborates the work of Riaud et al.[4, 5], and is further supported
by Quesnel et al.[6] where deep learning techniques were specifically employed
to successfully retrieve phase aberrations based on vortex coronagraphic
images.

We also note that numerical simulations can be misleading by generating non
random numerical artifacts: they may, for example, artificially resolve
ambiguities in pure radial modes, which is theoretically not expected. In this
respect, our analytical approach may provide a more reliable ground-truth in
the low aberration regime.

Real-world systems introduce additional complexities, including more intricate
pupil designs, the use of downsized Lyot stops, and the inherent limitations
of the vortex coronagraph, such as the leakage term. These factors can
potentially constrain the accuracy of phase retrieval, and should be
considered when applying phase retrieval techniques in high contrast imaging
systems with vortex coronagraphs.

## References

  * [1] Mathar, R. J., “Zernike Basis to Cartesian Transformations,” Serbian Astronomical Journal 179, 107–120 (Dec. 2009).
  * [2] Huby, E., Baudoz, P., Mawet, D., and Absil, O., “Post-coronagraphic tip-tilt sensing for vortex phase masks: The QACITS technique,” A&A 584, A74 (Dec. 2015).
  * [3] Niu, K. and Tian, C., “Zernike polynomials and their applications,” Journal of Optics 24, 123001 (Dec. 2022).
  * [4] Riaud, P., Mawet, D., and Magette, A., “Instantaneous phase retrieval with the vector vortex coronagraph. Theoretical and optical implementation,” A&A 545, A151 (Sept. 2012).
  * [5] Riaud, P., Mawet, D., and Magette, A., “Nijboer-Zernike phase retrieval for high contrast imaging. Principle, on-sky demonstration with NACO, and perspectives in vector vortex coronagraphy,” A&A 545, A150 (Sept. 2012).
  * [6] Quesnel, M., Orban de Xivry, G., Louppe, G., and Absil, O., “A deep learning approach for focal-plane wavefront sensing using vortex phase diversity,” A&A 668, A36 (Dec. 2022).