Wavefront sensing of optical vortices and complex wavefronts
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Wavefront sensing of optical vortices and complex wavefronts
Tengfei Wu1,2,3 , Pascal Berto2,3 , and Marc Guillon1,3,4,*
1
Université de Paris, SPPIN – Saints-Pères Paris Institute for Neurosciences, CNRS, 75006 Paris, France
2
Sorbonne Université, CNRS, INSERM, Institut de la Vision, 17 Rue Moreau, 75012 Paris, France
3
Université de Paris, 75006 Paris, France
4
Institut Universitaire de France (IUF), Paris, France
*
marc.guillon@u-paris.fr
arXiv:2101.07114v1 [physics.optics] 18 Jan 2021
Wavefront sensing is a non-interferometric, single-shot, and quantitative technique providing the spatial-
phase of a beam. The phase is obtained by integrating the gradient of the wavefront. This integration step is
especially delicate in the presence of optical vortices, which are topologically stable singular structures that
spontaneously appear in wavefields, and that are associated with non-conservative gradient-maps. Unlike
incomplete reconstructions typically achieved in such cases, we demonstrate a systematic approach for high-
resolution wavefront sensing of complex wavefronts. The method consists in applying an image processing
algorithm to the Helmholtz decomposition of the wavevector field. This improvement is expected to benefit
to several fields ranging from diffraction tomography to adaptive optics in scattering media.
Wavefront sensors (WFS) are simple and efficient devices modulation with negligible inter-pixel cross-talk [13], unlike
measuring wavefront (WF) distortions of optical fields. They continuous-surface deformable mirrors. Although optical vor-
have been used in many applications ranging from optical tices are detected by WFS [11, 14, 15, 16], the problem of
metrology [1] to spectroscopy [2]. Unlike interferometric the vortex-gradients integration provided by WFS has only
methods that provide a direct measurement of the phase, been achieved in simple cases using complicated pixel-based
WFS only measures phase gradients, i.e. the transverse com- reconstruction algorithms [17, 18]. To the best of our knowl-
ponent of the local wavevector. A numerical integration step edge, the systematic phase pattern reconstruction of complex
then rebuilds the WF. Advantageously, no reference arm is wavefields by WFS has not been tackled. The problem to
needed, not only providing higher stability and ease of im- rebuild singular WFs is mostly threefold. First, the phase
plementation but also making WFS compatible with inco- gradient map g measured by a WFS is a non-conservative
herent beams such as may arise from "guide stars". WFS vector field defined over a multiply connected domain. Di-
are thus invaluable tools for measuring aberrations in astron- rect spatial integration is thus not possible since the integral
omy [3], in ophthalmology [4] and for in-depth tissue imag- value depends on the line-path. Second, optical vortices are
ing [5, 6]. Nowadays, high-resolution WFS can achieve quan- associated with a infinite phase-gradient g at the singular-
titative phase imaging [7, 8] and thus represents an interest- ity locations, which appears as critically incompatible with
ing alternative to digital holography. Despite their poten- WFS. Third, WFS may imprecisely measure strongly fluc-
tial to measure complex WFs, WFS have only been used, so tuating gradients around anisotropic vortices, leading to in-
far, for measuring smooth distortions, such as optical aber- accurate vortex characterization. Thus, current WF recon-
rations (typically projected onto Zernike polynomials [9]) or struction techniques ignore what Fried has called the “hidden
optical-path-length profiles of thin and transparent biological phase” [19] and typically lead to an incomplete reconstruction.
samples [7, 8]. The main difficulty with complex wavefields Here, we propose a rigorous and complete WF reconstruction
– such as obtained when propagating coherent light through based on the Helmholtz’s and Stokes theorem.
random scattering media – arises from the numerous intrin- According to Helmholtz’s theorem [20], the vector field g
sic phase singularities, namely spiral phase dislocations (or can be split into an irrotational (curl-free) component and
optical vortices) of topological charge one [10]. a solenoidal (divergence-free or rotational) component [19,
The problem of phase-spirals integration has appeared 21, 12, 22]. All current integration techniques for WFS basi-
since the early ages of adaptive optics in astronomy [11]. cally consist in computing ∇ · g, so implicitly canceling the
In this context, it has been shown that neglecting branch- solenoidal component of the vector field. In WF shaping ex-
cuts significantly degrades adaptive-optics performances [12]. periments, neglecting a single optical vortex is equivalent to
Nowadays, high-resolution segmented spatial light modula- adding a complementary spiral phase mask, which has been
tors have been developped and easily allow spiral phase described as yielding a two-dimensional Hilbert transform of
1a) Addressed c) Irrota�onal WF d) Irr. WF + vor�ces phase ϕir and the solenoidal contribution of a vector potential
A. The sought-for complete phase profile ϕ, whose gradient-
field is g, can then be written as ϕ = ϕir + ϕs , where the sin-
gular phase contribution ϕs (or “hidden phase” [19]), defined
over a multiply-connected domain, satisfies ∇ϕs = ∇ × A.
Solving this latter equation then allows proper reconstruction
of the WF (Fig. 1d).
0
We now detail the steps allowing us to achieve rigorous
b) HD. First, the unicity of the HD both requires taking into ac-
count the contribution of a so-called additional “harmonic” (or
translation) term h, which is both curl-free and divergence-
free [20], and setting boundary conditions. The translation
SLM L1 L2 WFS term h accounts for the global tip/tilt of the WF and can
Fig. 1: Experimental vortex WF sensing. A phase pro- be conveniently included in the curl-free component (∇ϕir )
file containing optical vortices (a) is addressed onto a spa- by symetrizing the phase gradient field [27]. In addition,
tial light modulator (SLM) (b) illuminated by a colimated the unicity of the solution is further ensured by implicit
laser beam and imaged onto a high-resolution wavefront sen- periodic boundary conditions applied by computing deriva-
sor (WFS) with a telescope (L1 , L2 ). Considering only the tions and integrations through discrete Fourier transforms.
irrotational component of the gradient field detected by the Second, the potential vector A is solution of the equation:
WFS leads to an erroneous WF lacking optical vortices (c), ∇ × g = ∇(∇ · A) − ∆A. Determining the potential vector
unlike full Helmholtz decomposition of the gradient field (d). thus requires fixing a gauge. The Coulomb gauge ∇ · A = 0 is
chosen here for obvious convenience [19, 21, 12]. Since g is a
two-dimensional vector field (in say x, y plane), we then may
the field [23]. For complex speckled wavefields, such a sin- write A = Az ez without loss of generality. Third, noticing
gle spiral transform induces a major change in patterns since that gx + igy = (∂x + i∂y )(ϕir + iAz ), the HD ((1)) can be
resulting in an inversion of intensity contrasts [24, 25]. A vor- efficiently achieved numerically in the complex plane thanks
tex is essentially characterized by the circulation of the vector to a single computation step by projecting vectors onto the
flow around the singularity, namely its topologicalI charge (or circular unit vector σ+ = ex + iey :
winding number) n defined according to 2πn = g · d``. For
F [g · σ+ ]
C
optical vortices spontaneously appearing in freely propagat- ϕir − iAz = F −1
(2)
ik · σ+
ing beams, this topological charge has the specificity to be
an integer. Our reconstruction algorithm thus includes this where k stands for the two-dimentional coordinate vector in
quantization prior. Using a high resolution WFS previously the reciprocal Fourier space. The regular phase component
developed in our group [8], we then demonstrate experimen- ϕir is then recovered in a similar way as usually performed [26]
tally the possibility to rebuild complex WFs containing optical (Fig. 2b). The divergence-free component requires futher pro-
vortices of any topological charge. cessing steps to obtain the singular phase pattern ϕs from the
By way of illustration, a phase pattern exhibiting optical potential-vector component Az , as detailed hereafter.
vortices has been designed (Fig. 1a) and addressed to a phase- Let us first consider an optical vortex of topological
only spatial light modulator (SLM) (Hamamatsu, LCOS- charge n. Applying Stokes’ theorem to the definition of the
X10468-01), illuminated by a spatially filtered, polarized, and topological charge yields:
collimated laser beam (Fig. 1b). The SLM allows displaying I Z
patterns exhibiting both smooth local WF distortions (such 2πn = g · d`` = − ∆Az dS (3)
as lenses for the eyes and the face contour for instance) as C S
well as optical vortices of any topological charge (left- and
Reducing the contour length (and so the enclosed surface) to
right-handed optical vortices at the tips of the wiggling mus-
zero, it appears that −Az /(2πn) is the Green function of the
tache). A high-resolution wavefront sensor (WFS) [8], con-
two-dimensional Laplace equation. In theory, −∆Az /(2πn)
jugated to the SLM with a Galileo telescope in a 4 − f con-
is thus a Dirac distribution, making it easy to identify optical
figuration, then detects the phase gradient map of the WF.
vortices [19, 21] (see Fig. 2c). In principle, the corresponding
Direct numerical integration [26] then yields the regular WF
sought-for singular phase component ϕs could then be simply
shown in Fig. 1c, missing phase singularities because the non-
obtained by convoluting −∆Az /(2πn) by a single +1 optical
conservative (solenoidal) contribution to the WF-gradient has
vortex. However, in practice, rebuilding ϕs this way yields
been ignored. The full HD of the gradient vector-field can be
very poor results. The main difficulty is that the experimen-
achieved according to
tal −∆Az /(2πn) map is not a perfect Dirac distribution (or
g = ∇ϕir + ∇ × A (1) a single-pixeled non-zero data map): first because experimen-
tal data are affected by noise, and second, more critically,
splitting appart the irrotational contribution of the regular because they are filtered by the optical transfer function of
2a) b) curl-free component d) rebuilt phase
integration
c) superlocalized convolution
(filtered) segmentation
↳weighted charge map
centroid
↳topological 1
charge
0
solenoidal component
Fig. 2: Principle of full WF reconstruction. The divergence and the curl of the phase gradient map g (a) are computed
to extract the irrotational phase ϕir (b) and the solenoidal phase ϕs (c). Double space integration of ∇·g yields the irrotational
phase (a parabola in (b)). The curl of g yields −∆A where A is the potential vector of the Helmholtz decomposition. Image
segmentation and weighted centroid computation of the peaks in −∆A then allows reconstructing a Dirac-like distribution
whose convolution by a spiral phase profile yields ϕs . The complete phase ϕ is finally rebuilt by summing the two components
ϕir and ϕs (d).
vortex b) 10
charge -10 -5 -2 1 3 10
a) 5 slope=1.03
0
c) -5
rebuilt
phase -10
-10 -5 -2 0 1 3 10
vortex charge
Fig. 3: Reconstruction of phase spirals of various topological charges n from −10 to +10. The potential vector
A of a spiral phase of topological charge 1 is the Green function of Laplace equation. ∆Az thus exhibits a peak at the vortex
location (a). After segmentation, the integral computation of this peak yields the expected value 2πn (with a 3% precision)
(b). Phase profiles are then rebuilt (c).
the WFS. As detailed in Ref. [8], the optical transfer function a large enough surface S (enclosed by the contour) provides
of a WFS is especially limited by the non-overlapping con- the proper charge, under the Stokes’ theorem ((3)).
dition, which imposes a maximum magnitude for the eigen- To calculate numerically the location and the charge of op-
values of the Jacobi matrix of g (i.e. the Hessian matrix of tical vortices from the computed ∆Az map over the proper
ϕ). As a first consequence, the large curvatures of the phase surface areas, an image processing step is thus required, sum-
component ϕr (i.e. its second derivatives) may be underesti- marized in (Fig. 2c). The ∆Az map is first segmented using a
mated [8]. As a second consequence, the diverging magnitude watershed operation. Integration and weighted-centroid com-
(as 1/r) of g at an optical vortex location cannot be prop- putation over each segmented regions yields the charge and
erly measured, either the Hessian coefficients ∂x gy and ∂y gx . the precise location of each vortex. Second, a Dirac-like vortex
Therfore, the measurement of the vector potential Az is wrong map is rebuilt based on the result of the former step, and con-
in the vicinity of the vortex center and the obtained peak is voluted by a +1 spiral phase mask to yield ϕs (see Fig. 2c).
not single-pixeled (Fig. 2c). Nevertheless, the circulation of Finally, the complete phase reconstruction ϕ = ϕir + ϕs is
g in (3) can yield an accurate measure of the vortex-charge computed and wrapped between 0 and 2π (Fig. 2d). In our
provided that the contour is chosen at a large enough distance specific experimental implementation, noise was reduced by
from the vortex center, where g is accurately measured by the filtering −∆Az using a Gaussian function whose width was
WFS. Consequently, although the estimate of ∆Az is wrong set according to an estimate of the WFS resolution [8] (10
in the vicinity of the vortex, the peak integral achieved over camera-pixels). This step avoided oversegmentation, so im-
3proving charge-measurement reliability, and speeding up the a) Addressed to the SLM b) Rebuilt at the WFS
processing time. The latter was measured to be 0.54s on an
Intel® Core i5-9400H CPU for a 1.3 Mpx map at maximal
vortex density.
To demonstrate the efficiency of this approach to charac-
terize and rebuild optical vortices, we addressed phase spi-
rals with charges ranging from −10 to +10 (Fig. 3). Be-
cause of the diverging phase gradient at the vortex center,
the ∆Az maps exhibit peaks whose widths increase with the c) Phase error d)
charge of the vortex n (Fig. 3a). Nevertheless, integration of
−∆Az /(2π) over segmented regions yields n within a 3% ac-
curacy range (Fig. 3b). Rounding the integral to the closest
integer value allowed an accurate reconstruction of the opti-
cal vortices (Fig. 3c). Differences between the rebuilt phase
profiles and the perfect ones addressed to the SLM are due to
the contribution of ϕir arising from uniformity imperfections
of the SLM.
Finally, we demonstrate the possibility to retrieve the Fig. 4: WF reconstruction of a complex wavefield.
phase of complex random wavefields. Random wavefields Equiphase-line structures 0 - 2π visible in the phase pattern
contain a high density of optical vortices of charge +1 and addressed to the SLM (a) are clearly recovered in the rebuilt
−1 [28, 10, 29]. These vortices exhibit elliptical phase and phase pattern (b). A few of these equiphase-lines are high-
intensity profiles along the azimuthal coordinate. The non- lighted in green. Slight discrepancies are observed (c) due to
uniform increase of the phase around the singular point may a low-spatial-frequency component of ϕir . The ∆Az map al-
then alter the ability to detect them if the phase-gradient lows a clear identification of the charge and the location of
magnitude is locally too large. Furthermore, the separa- optical vortices.
tion distance between vortices may be much smaller than the
speckle grain size, especially when close to creation or anni-
hilation events of pairs of vortices [30, 31]. Such a complex tribution of a so-called additional “harmonic” (or translation)
wavefield was numerically generated by taking the Fourier term h, which is both curl-free and divergence-free [20]. In
transform of a random phase map of finite aperture, and ad- practice, the translation term h, accounting for tilted WFs,
dressed to the SLM (Fig. 4a). Despite the aforementioned spe- can be included in the curl-free component (∇ϕir ), which
cific difficulties, the WF could be efficiently rebuilt (Fig. 4b). we achieve here thanks to symetrization operations on the
To underline so, we materialized some of the 0-2π equiphase phase gradient map [27]. Unicity of the solution is further
lines of the input phase in green. These equiphase lines are ensured by implicit periodic boundary conditions applied by
easy to identify with a gray-level colormap because of the achieving derivation and integration computations through
abrupt white-to-black drop. The difference between the re- discrete Fourier transforms. Full reconstruction of WFs with
built and the input phase profiles is shown in Fig. 4c demon- a WFS represents a important step to make WFS performant
strating the almost perfect identification of optical vortices reference-less phase detectors and to allow random wavefields
(a single strongly elliptical vortex was missed at the bottom characterization with incoherent light sources. These develop-
right of the image). Again, differences mostly appear on the ments are of interest for applications such as adaptive optics,
ϕir contribution on the edges of the SLM, where the SLM diffractive tomography, as well as beam shaping behind scat-
reliability degrades. The dense experimental map of vortices tering and complex media.
distribution and charges is shown in Fig 4d.
Relying on a high-resolution WFS, we could thus propose
a systematic and robust approach to rebuild optical vortices Funding Information
of various charges as well as complex random WFs contain-
ing optical vortices. The proposed method first consists in This work was partially funded by the french Agence Na-
performing a HD of the local wavevector field g measured by tionale pour la Recherche (SpeckleSTED ANR-18-CE42-0008-
the WFS. The systematic reconstruction of the optical vortex 01) and by the technology transfer office SATT/Erganeo
map is achieved thanks to image processing steps. Impor- (Project 520).
tantly, the circulation of g/(2π) over vortices, computed as
the integral ∇ × g over large enough surface areas (under the
Stokes theorem), yields the topological charge of vortices. The Acknowledgments
robustness of phase-spiral-reconstructions further relies on the
quantization prior about the detected topological charges. The authors thank Jacques Boutet de Monvel and Pierre Bon
Noteworthy, in principle, the unicity of the HD both requires for careful reading of the manuscript, and Benoit Forget for
setting boundary conditions and taking into account the con- stimulating discussions.
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