# High-definition quadriwave lateral shearing interferometry

Benoit Wattellier, Anaïs Saintoyant, Julien Savatier, Lucie de Laulanie, Sherazade Aknoun, Roman Zinchuk, Serge Monneret

► To cite this version:

Benoit Wattellier, Anaïs Saintoyant, Julien Savatier, Lucie de Laulanie, Sherazade Aknoun, et al. High-definition quadriwave lateral shearing interferometry. Journal of the Optical Society of America. A Optics, Image Science, and Vision, 2024, 41 (11), pp.C99. 10.1364/JOSAA.533811. hal-04790523

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Submitted on 27 Mar 2025

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High Definition Quadriwave Lateral Shearing Interferometry

BENOIT WATTELLIER,¹,* ANAÏS SAINTOYANT,¹ JULIEN SAVATIER,² LUCIE DE LAULANIE,¹ SHERAZADE AKNOUN,¹ ROMAN ZINCHUK,¹,² AND SERGE MONNERET²

¹ PHASICS, Parc Technologique de Saint-Aubin, 91190 Saint-Aubin, FRANCE

² Aix Marseille Univ, CNRS, Centrale Med, Institut Fresnel, Marseille, FRANCE

*Corresponding author: bw@phasics.com

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX

We present an evolution of quadriwave lateral shearing interferometry (QLSI) that improves the definition of quantitative phase images. It is now possible to produce images with as many pixels as the camera that records the interferogram. This is done by moving a diffraction grating in front of the camera and linearly combining at least 9 acquisitions. In this article, we present the principle of this technique and illustrate it by several examples acquired on a microscope with both calibrated and biological samples. We demonstrate the possibility to produce quantitative phase images with 5.5 million of pixels, which is to our knowledge the largest images ever produced by a wave front sensor.

Introduction

Because living cells are mostly made of water, they generally
do not absorb much visible light. Consequently, bright-field
images often exhibit low contrast, which decreases further as
magnification increases. Several techniques have been
proposed to improve the contrast of microscopic images, with
sample dyeing and fluorescence labeling being the most
commonly used solutions. However, these methods require
altering or even killing the living objects. To overcome this
limitation, Fritz Zernike proposed utilizing the light phase
information to create contrast in label-free images [1]. The
light phase shift introduced by an object is strongly related to
the time photons take to propagate through it. In semi-
transparent media, where scattering can be neglected, the
phase is the accumulation of the refractive index during
propagation. This reveals mass density inhomogeneities in
biological samples.

For a long time, phase information was only used to create
contrast in unlabeled samples. The advent of digital
interferometry allowed for the quantification of phase shifts,
ushering in the era of quantitative phase imaging (QPI), which
explores beyond mere contrast. A main application of
quantitative information in QPI is dry mass measurement.
Barer [2] demonstrated that the refractive index of a solution
is proportional to its solute mass density, with the ratio being
nearly constant for most biological molecules. Dry mass is
calculated by integrating the phase shift between a cell and its
extracellular environment over its surface. QPI is used to
monitor cell metabolism and cell cycle [3] [4] [5], dry mass
transport [6], assess cytotoxicity [7] [8], analyze tumor cell
responses to drugs [9] and more generally study biomedecine
[10] [11], and biophysical processes involved in the cell shape

and volume regulation [12]. It has also been used to resolve
neuronal network activities [13] or to study transmembrane
water flux regulation in cells [14]. Temperature changes at the
microscopic level are also measured with a precision of less
than 1 K [15], enabling studies on hyperthermal bacteria or
archaea [16].

The first implementation of quantitative phase imaging was based on interferometry and is known as digital holography, an adaptation of Mach-Zender interferometry to microscopy [17]. Since then, a lot of techniques have been proposed, including DPM [18], SLIM [19], ptychography [20] or ITE equation solving. 3D refractive index recovery is also possible by combining the fields diffracted by the sample from several plane wave illumination angles [21] [22] [23].

In 2009, Bon et al. [24] proposed using a wavefront sensor
to measure the complex transmittance of a semi-transparent
object under a microscope. Wavefront sensors measure the
complex field of an impinging light beam and are primarily
used in optical and laser metrology as well as adaptive optics.
Historically, their lack of resolution made them unsuitable for
producing phase images. However, since Jérôme Primot
invented Quadriwave Lateral Shearing Interferometry (QLSI)
[25], the spatial resolution of wavefront sensors has improved
by a factor of 4, and the number of pixels per image by a factor
of 16. Wavefront sensors are now off-the-shelf, industrialized,
and accurate instruments that interface like standard cameras.
They can be attached to any microscope exit port and handle
any kind of light source in the visible and near-infrared
spectral ranges.

Despite these advantages, the spatial resolution of QLSI
limits its applications in microscopic imaging. It is sometimes
considered inferior to sharp fluorescence images and
struggles to resolve details such as small organelles (vesicles)
or cytoskeleton fibers. This relatively low resolution is due to
the requirement of between 3x3 and 4x4 camera pixels to
recover a single phase and intensity pixel, resulting in a
resolution that is one-fourth of the camera's.

In this paper, we propose an evolution of QLSI called
High-Definition QLSI (HD-QLSI), which enables a definition
of 1. This means that quantitative phase images have the same
number of pixels as the camera acquiring the interferogram
data. The reason why QLSI needs 4x4 camera pixels can be
explained using either direct or Fourier space approaches,
with the former being more intuitive and the latter more
rigorous.

Using a direct approach, we return to the origins of QLSI,
which is the Hartmann test. Light is sampled by a grid of
holes, and the signal observed a few millimeters behind the grid is a projection of the holes. The positions of these projections change with the local light ray direction. The projected hole positions are spotted and compared to an ideal situation where all the rays propagate in the same direction, indicating phase gradients. These phase gradients are numerically integrated to produce the phase image. The necessary separation of the holes requires many camera pixels per phase pixel, depending on the sampling grid technology. In short, we obtain the field value for any pixel located behind each hole of the grid. However, we do not have the information about the phase at the surrounding pixels, since light is blocked by the grid.

The simple idea behind HD-QLSI is that we move the sampling grid with respect to the camera chip, so that we obtain phase values at other pixels. This approach eventually achieves phase images with the same spatial resolution as the camera, from as many acquisitions as the number of pixels in one grating period. However we will show that a wave optics approach reduces the number of necessary moves to cover all the camera pixels. In this case, the phase derivatives are recovered by a simple linear combination of the different images. In the end, only 9 moves are necessary to improve the number of phase pixels by a factor of 16.

This paper details the principles of HD-QLSI briefly exposed here, provides an experimental implementation, and characterizes its improvements in spatial resolution and dynamic range while maintaining the metrological assets of standard QLSI. Finally, we illustrate the benefits of HD-QLSI for QPI in biological imaging.

## High definition QLSI principle

We describe the incident complex light field as

$$A(x,y) = \sqrt{I_0(x,y)} \exp j\left(\frac{2\pi}{\lambda} W(x,y)\right) \quad (1)$$

As recalled later, QLSI does not depend on the wavelength of the incident field, which leads us to favour the so-called Optical Path Difference (OPD) as the output of the technique, written W (for Wavefront) here. Often expressed in unit of nanometers, W or OPD is simply related to the phase (or argument) of the complex field:

$$\varphi = \frac{2\pi}{\lambda} W \quad (2)$$

We prefer to use the OPD for microscopy and biological applications because it does not depend on the light source used to measure samples mostly considered as achromatic. Moreover dry-mass experiments, which are of great importance in QPI for biology, are based on the OPD, thanks to the direct conversion of the OPD into a dry-mass surface density using the relation inspired by Barer studies [2].

## Quadriwave Lateral Shearing Interferometry and Multi-Lateral Shearing Interferometry

In Quadriwave Lateral Shearing Interferometry (QLSI), the electromagnetic field impinges a diffraction grating, so-called Modified Hartmann Mask (MHM) [25]. This consists in the combination of an intensity Hartmann mask of period p superimposed with a π-shift phase mask with a 2p period. It

replicates the field in at least four identical diffraction orders. These orders are slightly tilted with respect to the optical axis. They propagate towards a camera sensor located at a distance z, namely a few millimeters from the grating, where they interfere. It was previously shown [26] that the fringe pattern deformation is proportional to the OPD (or phase) derivatives.

The interferogram intensity $I_C$ can be seen as the sum of a set of diffraction order couple interference intensities. If diffraction is neglected (small distance from the grating to the camera sensor), each couple codes for a phase derivative in one direction as seen in Eq. 3 and [24].

$$I_C(x,y) = I_0(x,y) \left\{ \begin{aligned} & \exp j\left(\frac{2\pi}{p}x + \frac{1+2\pi z}{p} \frac{\partial W}{\partial x}(x,y)\right) + \\ & \exp j\left(\frac{2\pi}{p}y + \frac{1+2\pi z}{p} \frac{\partial W}{\partial y}(x,y)\right) + \\ & \exp j\left(\frac{2\pi}{p}(x+y) + \frac{2\pi}{p}z \frac{\partial W}{\partial(x+y)}(x,y)\right) + \\ & \exp j\left(\frac{2\pi}{p}(x-y) + \frac{2\pi}{p}z \frac{\partial W}{\partial(x-y)}(x,y)\right) + \\ & \qquad + c.c. \end{aligned} \right\} \quad (3)$$

The intensity is made of 9 terms, which carry information about the intensity and the phase of the incident field. Each term is associated to a unique spatial frequency or harmonic. For instance, the derivative along x, $\frac{\partial W}{\partial x}(x,y)$, is linked to the modulation $\exp j(\frac{2\pi}{p}x)$ and the harmonic located at the position $(\frac{1}{p}, 0)$ in the Fourier space.

The expression of the interferogram shows that the wavelength dependences upon diffraction and upon interference are compensated. Consequently, the interferogram does not depend upon the wavelength, which allows using this technique with any wavelength without a specific calibration and even with white light, as previously discussed in several publications such as [24]. Measurement resolution (noise) and dynamic range can be adjusted by changing the distance z between the grating and the sensor.

In a more general case, not only four orders of diffraction are generated. For instance, if the light wavelength differs from the grating design wavelength – the wavelength where the phase-shift is nominally equal to π – a zero-order appears which interferes with the four first orders. This adds 4 terms to the equation, which now contains 9+4=13 spatial frequencies. If the geometry is not square any more but remains periodic, more diffraction orders are to be considered. But in any case, the interferogram intensity is made of N terms corresponding to N spatial frequencies. Using such diffractive optics to measure an electromagnetic field is called Multi-Lateral Shearing Interferometry (MLSI). It includes wave front sensing techniques known as Hartmann test or Shack-Hartmann. We will show how our High Definition evolution seamlessly applies to such techniques, at the expense of more displacements and a longer measurement time.

## Standard phase and intensity recovery method from a QLSI interferogram

We have just shown that the intensity recorded by a camera sensor is the sum of 9 terms containing the incident field intensity and phase information and associated to specific
spatial frequencies. The aim of the field recovery process is to solve the equation for $I_0(x,y)$ and $W(x,y)$, knowing $I_C(x,y)$. The standard solution proposes to do so from a unique camera acquisition at the expense of a strong assumption on the electromagnetic field spatial frequency support.

Let us consider the Fourier transform of the interferogram values. It is made of 9 peaks or harmonics which are convoluted by functions of the intensity and phase (see Figure 1).

$$ 
\begin{aligned}
FT(I_C) = & FT(I_0) + \delta(\frac{1}{p},0) \otimes FT\{I_0 \exp j(\frac{2\pi}{p}z \frac{\partial W}{\partial x})\} + \\
& \delta(0,\frac{1}{p}) \otimes FT\{I_0 \exp j(\frac{2\pi}{p}z \frac{\partial W}{\partial y})\} + \\
& \delta(\frac{1}{p},\frac{1}{p}) \otimes FT\{I_0 \exp j(\frac{2\pi}{p}z \frac{\partial W}{\partial(x+y)})\} + \\
& \delta(\frac{1}{p},-\frac{1}{p}) \otimes FT\{I_0 \exp j(\frac{2\pi}{p}z \frac{\partial W}{\partial(x-y)})\} + \text{c.c.}
\end{aligned}
\quad (4) $$

If we assume that the frequency support of these functions is smaller than the distance between the peaks, we can separate each function by spectrally windowing the Fourier transform around the 9 peaks. After an inverse Fourier transform, we obtain functions of the intensity and phase derivatives in 9 directions. The phase derivatives are finally numerically integrated and lead to the electromagnetic complex field.

<figure><img src="image_2.png" /><figcaption>Figure 1: QLSI analysis with the Fourier transform of the interferogram at left. Windowing around the central harmonic H0 enables to compute wave front intensity while the phase gradients are computed with windowing around the first harmonic in both directions. Finally, a 2D integration enables to get the phase.</figcaption></figure>

One main limitation to QLSI for QPI is its definition. In a standard implementation, 16 pixels are required to make 1 pixel of phase. This loss of resolution originates from the spectral windowing necessary to discriminate intensity and phase modulations. Since we need to separate the different peaks and leave space between them, the frequency support for each peak is much less than the whole Fourier space. For instance, in the standard configuration, the grating period is set to be 4 times the camera pixel size. Therefore, the spacing between two peaks is limited to Nyquist/4, which means that field images have a definition 4x smaller than the camera: they have a definition of 1/4th. In other terms, 4x4 camera pixels are needed to calculate one phase pixel. If the field actually contains spatial frequencies higher than the distance between two peaks, the peaks overlap and information cross-talks leading to artefacts, known as Moiré effect.

However, this standard recovery method allows single-shot field measurements which open applications based on short-pulsed lasers or fast phenomena observation [27].

In the following, the aim of the study is to show how a new HD-QLSI method is a way to overcome these limitations on definition and sampling. We now focus on the theoretical derivation of this method.

## Principle of spatial phase shifting

In the introduction, we proposed to move the grating in front of the camera sensor in order to probe the beam wave front at various positions within a grating period. One way to recover the phase images would be to interlace the acquisitions pixel by pixel. Not only this would be complex to be done accurately but, as we will show now, this is not the most efficient way to exploit the gathered information. We will take benefit of the periodicities of the interferograms: a translation of a sinusoidal function is equivalent to the addition of a phase term, which is proportional to the function frequency. We will now show that a more efficient way to process the acquired interferograms is to linearly combine them.

Let us begin with a simple version of the problem to be solved by considering equation 3 in a 1D configuration. It can be rewritten as

$$ I(x) = I_0(x) \left\{ 1 + \cos\left(\frac{2\pi}{p}(x) + G(x)\right) \right\} \quad (5) $$

Where the unknown functions $I_0(x)$ and $G(x)$ are entangled.

When we move the grating with respect to the camera by a distance $dx_k$, keeping everything else still, the interferogram intensity $I_k$ is

$$ I_k(x) = I_0(x) \left\{ 1 + \cos\left(\frac{2\pi}{p}(x + dx_k) + G(x)\right) \right\} \quad (6) $$

It is important to notice that each measurement only differs by the phase-shift introduced by the displacement $dx_k$. $I_0(x)$ and $G(x)$ are measured at the same location. If we develop this expression, we obtain

$$ 
\begin{aligned}
I_k(x) = I_0(x) &+ \cos\left(\frac{2\pi}{p}(x + dx_k)\right) I_0(x)\cos(G(x)) \\
&-\sin\left(\frac{2\pi}{p}(x + dx_k)\right) I_0(x)\sin(G(x))
\end{aligned}
\quad (7) $$

This can be rewritten in a matrix form for each position $x$, as

$$ I_k = M_{kl}G_l \quad (8) $$

where

$$ I_k = I_k(x) $$

$$ M_{kl}(x) = \begin{bmatrix} 1 & c_1(x) & s_1(x) \\ 1 & c_2(x) & s_2(x) \\ \vdots & \vdots & \vdots \\ 1 & c_n(x) & s_n(x) \end{bmatrix} \quad \text{with} $$

$$ c_k(x) = \cos\left(\frac{2\pi}{p}(x + dx_k)\right) \quad \text{and} $$

$$ s_k(x) = \sin\left(\frac{2\pi}{p}(x + dx_k)\right) $$

$$ G_l(x) = \begin{bmatrix} I_0(x) \\ I_0(x)\cos(G(x)) \\ I_0(x)\sin(G(x)) \end{bmatrix} \quad \text{is the unknown information} $$

The vector $G_l(x)$ is recovered from the $I_k(x)$ measurements using the pseudo-inverse of $M_{kl}(x)$ so that

$$ G_l = M_{kl}^{-1} I_k \quad (9) $$
In the end, the function $G(x)$ is deduced from the 2nd and 3rd components of $G_l(x)$

$$G(x) = \arg\{G_3 + iG_2\} \qquad (10)$$

We managed to recover the information coded in a frequency modulation by phase-shifting the periodic part of the information. We can generalize this principle to the situation where the function $I(\mathbf{r})$ is a linear combination of periodic functions at known frequencies $u_l$.

$$I(\mathbf{r}) = \sum_{l=1}^{N} G_l(\mathbf{r}) \exp(j\mathbf{u}_l \cdot \mathbf{r}) \qquad (11)$$

We assume that we are able to make $k$ measurements $I_k(\mathbf{r})$

$$I_k(\mathbf{r}) = \sum_{l=1}^{N} G_l(\mathbf{r}) \exp(j\mathbf{u}_l \times (\mathbf{r} + d\mathbf{r}_k)) \qquad (12)$$

In order to recover the information $G_l(\mathbf{r})$, we follow the same path as before and introduce

$$M_{kl}(\mathbf{r}) = \exp(j\mathbf{u}_l \times (\mathbf{r} + d\mathbf{r}_k)) \qquad (13)$$

The functions $G_l(\mathbf{r})$ are recovered from the inverse or pseudo-inverse of $M_{kl}(\mathbf{r})$

$$G_l = M_{kl}^{-1} I_k \qquad (14)$$

Like in the first case, we show that it is possible to recover information encoded in frequency modulation by phase-shifting the periodic part of the signal.

## Application to QLSI

We now apply the principle of spatial phase shifting to Quadriwave Lateral Shearing Interferometry. If we observe equation (3), we identify a situation like in the previous section with the $G$ vector being

$$G_l(\mathbf{r}) = I_0(\mathbf{r}) \begin{bmatrix} \exp j \frac{2\pi}{p} z \frac{\partial W}{\partial x}(\mathbf{r}) \\ \exp j \frac{2\pi}{p} z \frac{\partial W}{\partial y}(\mathbf{r}) \\ \exp j \frac{2\pi}{p} z \frac{\partial W}{\partial y}(\mathbf{r}) \\ \exp j \frac{2\pi}{p} z \frac{\partial (x+y)}{\partial W}(\mathbf{r}) \\ \exp j \frac{2\pi}{p} z \frac{\partial (x-y)}{\partial W}(\mathbf{r}) \end{bmatrix}_{\text{c.c.}} \qquad (15)$$

The expression for $I(\mathbf{r})$ corresponds to a given position of the 2D diffraction grating with respect to the camera. If we move alone the grating by an amount($d\mathbf{r}_k$) without moving the camera with respect to the beam being measured, we obtain $N$ intensity measurements $I_k(\mathbf{r})$, which expression is

$$I_k(\mathbf{r}) = M_{kl}(\mathbf{r}) G_l(\mathbf{r}) \qquad (16)$$

With the matrix $M_{kl}(x,y)$ defined from its columns

$$M_{kl}(\mathbf{r} = (x, y)) = \begin{bmatrix} \exp j \left( \frac{2\pi}{p} (x + x \cdot d\mathbf{r}_k) \right) \\ \exp j \left( \frac{2\pi}{p} (y + y \cdot d\mathbf{r}_k) \right) \\ \exp j \left( \frac{2\pi}{p} (x + y + (x + y) \cdot d\mathbf{r}_k) \right) \\ \exp j \left( \frac{2\pi}{p} (x - y + (x - y) \cdot d\mathbf{r}_k) \right) \end{bmatrix}_{\text{c.c.}}$$

Like for spatial phase-shifting the $G_l$ vector is recovered by multiplying the measurements by the inverse of the matrix $M_{kl}$.

In the case of QLSI, the vector $G_l$ has 9 components. If we make 9 measurements, the $M_{kl}$ matrix is square and invertible. If we choose to make more measurements, we need to use the pseudo-inverse of $M_{kl}$matrix. This reduces the measurement noise and, in some cases, gives access to the uncertainty on the measurement. Since the 9 components are not all independent, it could be possible to use less measurements. In this case, the method would be more complex because based on iterations.

This method has an interesting property with respect to the standard method to solve the QLSI equation: the gradient information is recovered only from a linear combination (a matrix-multiplication) of the intensity measurements. This is a computationally very efficient method, easily coded for GPU (Graphics Processing Unit, based on parallel computing). In the case of the standard method, at least 6 Fourier transforms are needed to recover the same information.

Moreover we do not need to compute all the 9 components of the $G_l$ vector, since the 3 first ones lead univocally to the phase and intensity information.

We can also combine all the 9 components to reduce measurement noise as explained in [28].

In this section, we showed that acquiring multiple interferograms with multiple lateral positions of the grating without moving the camera allows to recover the intensity and phase information of the incident beam for each pixel of the camera. This method is called High Definition QLSI (HD-QLSI). In this case, the definition of the measurement is 1.

## Generalization to HD-MLSI

The principle of HD-QLSI can be generalized to MLSI, which principle is very similar to QLSI. The only difference is the number of harmonics in the acquired interferograms and therefore the number of components in the $G_l$ vector. In particular, if we consider the zero-order of the diffraction grating, this vector now has 13 components. Therefore at least 13 acquisitions are necessary to recover the phase and intensity information.

## Experimental implementation

We implemented the principle of High Definition QLSI in a prototype that includes a sCMOS camera (Zyla 5.5, Andor, UK) comprising 5.5 Mpixels with 6.5µm size (see Figure 2). We used a standard MHM grating implemented in commercial QuadriWave Lateral Shearing interferometer (SID4Bio, Phasics SA, Saint Aubin) which has a period of 29.6µm. This grating is fixed on a nanometric stage designed for this application (nPoint, US) ensuring requested performances of displacement and repositioning. Simulations were performed to determine stage's parameters to optimize the use of the prototype. With the characteristics of the MHM used, the optimal value for a single displacement is about a dozen of micrometers following a 3x3 grid.
To optimize the phase measurement in terms of dynamics and sampling, the grating must be located at less than one millimeter from the sensor. As it is very close to the sCMOS sensor and in motion, for prototyping practical considerations, a telecentric lens with 1x magnification is used to image the grating in front of the sensor. This also allows to achieve arbitrary grating to sensor distance, and thus adjustable sensitivity.

The prototype produces 5.5x10⁶ pixels intensity and phase images, with a lateral resolution of 6.5µm. It can be integrated on any experimental setup as it is used as a standard image sensor. Here, we use an inverted Nikon microscope (TE2000U, Nikon). The source is a halogen source which spatial coherence is controlled by a Koehler system. We added a band-pass filter (either ca. 550 nm or 650 nm; bandwidth of ~50 nm, e.g. FF01-609/54-25 from Semrock) to minimize chromatic dispersion in the observed sample and in the telecentric lens imaging the grating onto the CCD sensor. We use different objectives (magnification 2.5x to 100x and NA from 0.075 to 1.3) (Nikon, Japan), which are mentioned in the following figure captions. The prototype is mounted on a C-mount adapter so as the detector plane matches the microscope image plane.

<figure><img src="image_3.png" /><figcaption>Figure 2: HD-QLSI prototype schematic and its connection to a microscope. (solid lines) light path for one source point. (dashed line) light rays diffracted by the sample and imaged onto the HD-QLSI sCMOS sensor (CAM). The moving grating is imaged a fraction of millimeters from the sCMOS sensor (blue dotted lines). A spectral filter is introduced between the microscope objective and the tube lens.</figcaption></figure>

## Qualification and performance

In this part, we study some of the metrological performances of the HD-QLSI prototype and compare it to standard QLSI. This includes accuracy, precision, dynamic range, acquisition speed and spatial resolution.

### Accuracy: noise and trueness

To assess the accuracy of wave front measurements with the HD-QLSI, we need to consider its precision and its trueness. Precision is the dispersion of values under repeatability conditions [29], whereas trueness is the

difference between the measurement and a reference value. This is the hardest to estimate since reference samples are often hard to find. In this study, we focused on noise (often identified to resolution) and trueness.

To measure the HD-QLSI noise, we acquire series of 20 phase images and study the value dispersion. We first subtract the average phase to all phase images. We then estimate the phase noise by two means. The spatial noise is derived by averaging over the acquisitions the phase standard deviation of each image. The temporal noise is the average over the entire image of the standard deviation of each pixel value. In the case of the HD prototype, the measured spatial noise is 0.68nm and the temporal noise is 0.74nm. This noise can be compared to QLSI noise observed with commercial products (SID4-sC8, PHASICS, France) which spatial noise is specified at 0.5 nm [3]. The slight increase is expected since the repositioning noise is the dominant noise in HD-QLSI phase images and the translation stage accuracy was set to generate a phase noise around 1 nm.

To assess the trueness of the HD-QLSI prototype, we use a calibrated etched slide (Arnano, FRANCE). This sample is made of sapphire ($n = 1.77@ 543nm$) and has different patterns engraved with a depth of 100nm ± 10nm. These are binary phase patterns. In this section, we focus on square patterns.

In the projection mode frame, the transmitted optical path difference (OPD) is linked to the local refractive index $n(x,y)$ and the etching depth $t$ thanks to the relation:

$$OPD(x,y) = \int_{0}^{t} [n(x,y) - 1] dz \quad (17)$$

Considering the refractive index of the Arnano slide, a phase shift of 77nm ± 8nm is expected The measured phase shifts both agree with the theory (see Figure 3). With HD-QLSI phase images, an OPd of 76.6 nm ± 1.0nm is measured whereas an OPD of 76.1nm ± 0.6nm is obtained with QLSI measurements.

HD measurements seem a little noisier inside the etched parts. This higher noise comes from two phenomena. As seen before, HD-QLSI images are slightly noisier than standard QLSI images. But HD-QLSI images have access to higher frequencies which are washed out by QLSI reduced sampling. Therefore, etching defects are better resolved and result in increased noise.

<figure><img src="image_4.png" /><figcaption>Figure 3: (left) HD-QLSI phase image of a pattern of Arnano slide at 40x, 0.6 NA imaging. (right) cross section along the black solid line. The same sample was also measured with a standard QLSI (image not shown here for sake of visibility), which cross-section at the same point is also given in orange.</figcaption></figure>

### Spatial resolution
In principle, HD-QLSI gives access to higher spatial frequencies than standard QLSI. In this part, we prove that it is true.

It is interesting to first check that the prototype spatial resolution matches the microscope images resolution. The HD-QLSI Nyquist frequency $f_{Ny}$, which correspond to the maximal frequency resolved by a sampling system, is linked to the sensor pixel size $pix$ by

$$f_{Ny} = \frac{1}{2.pix} \qquad (18)$$

For the prototype studied here which has a pixel size of 6.5µm, we have $f_{Ny} = 77 lp/mm$. For best results, this frequency should be greater or equal to the microscope image maximal spatial frequency $f_{Ny}$ considering incoherent illumination. This depends upon the wavelength $λ$, the objective numerical aperture $NA_{obj}$ and the microscope magnification $γ$

$$f_{cut-off} = \frac{2NA_{obj}}{\gamma\lambda} \qquad (19)$$

For instance, with a 10x objective with $NA_{obj} = 0.3$ at $λ = 550 nm$, $f_{cut-off} = 100 lp/mm$. It is the same for the 2.5x, NA=0.075 objective. For a 100x objective with $NA_{obj} = 1.3$ at $λ = 550 nm$, we obtain $f_{cut-off} = 47 lp/mm$. This means that 10x images will be slightly undersampled whereas 100x HD-QLSI images should be able to resolve any details in the sample which are transmitted by the microscope objective.

We first compare the resolution of QLSI and HD-QLSI images in the same configuration on a microscope. For this purpose, a phase-only USAF resolution test chart and a radial Siemens chart (Arnano, France) are imaged with the same objective (10x, 0.15 NA) (see Figure 4). We easily see that HD-QLSI images resolve higher frequencies than QLSI images. Please note that the highest pattern spatial frequencies were chosen to match the HD-QLSI resolution. This means that the QLSI wave front sensor was illuminated with spatial frequencies much higher than what it could resolve. This results in artefacts due to undersampling. For instance, Moiré phenomenon is visible with the formation of low frequencies at the center of the radial test target.

<figure><img src="image_5.png" /><figcaption>Figure 4: Phase images of USAF resolution test chart (top) and Siemens star (bottom) respectively obtained with QLSI (top and bottom left) and HD-QLSI phase image (top and bottom right) acquired in the same configuration on a microscope (10x objective, NA 0.15).</figcaption></figure>

To quantify the ability of HD-QLSI to resolve image high frequencies, we made experimental MTF measurements. Various methods are possible to do so. However few are easy to implement and interpret. We chose to use a Siemens star pattern which gives access to a lot of spatial frequencies at once. For these measurements we used the specifically designed calibration slide (Arnano, FRANCE) mentioned above. In particular we used the Siemens star pattern with 204 cycles and a height of 100nm. It covers a 1.3mm diameter's disk on the slide. This is made of 204 etched triangles, all of them pointing to the star center. A large number of spatial frequencies are encoded in this star: their modulus is given by the distance to the center whereas the direction is given by the azimuth. The engraving spatial resolution of these different patterns on the slide is one micrometer.

To retrieve the MTF for a specific frequency, we make azimuthal cross-section of the Siemens star image, which means we collect pixel values along circles centered on the star center. The radius of each circle is related to the spatial frequency. The contrast for each frequency is calculated from the azimuthal cross-section standard deviation (STD). If we consider a square pattern of period p and height a, the average value is $a/2$ and the square distance to this average is $(a/2)^2$ for any point in the pattern. This means that the cross-section value standard deviation is $a/2$. Since a is the sought contrast, it is easily extracted from the standard deviation of the azimuthal cross-sections. This method corrects any inhomogeneities due to substrate imperfections, such as the “white” areas seen on the HD Siemens image.

If we normalize this contrast to 1 for the lowest frequencies, we obtain a modulation transfert function that fits the expected values from theory. The theoretical values include the imaging system MTF.
<figure><img src="image_6.png" /><figcaption>Figure 5: MTF curve deduced from the imaging of a Siemens star target. (blue solid line) theoretical MTF (circles) experimental values. For comparison, we give the theoretical MTF for standard QLSI (orange dashed line)</figcaption></figure>

## Dynamic range

In addition to the gain in resolution, this technique leads to a gain in dynamic range. This is because the gradients are sampled by more points. To highlight this, different phase images of 20 µm diameter polystyrene beads in immersion oil were made at 10x magnification NA 0.15 with HD-QLSI and QLSI techniques (see Figure 6). Please note that we intentionally used magnification and samples that best fit the HD-QLSI dynamic range, leading to expected artefacts due to undersampling with QLSI.

<figure><img src="image_7.png" /><figcaption>Figure 6: Phase images of polystyrene beads at 10x magnification NA 0.15, measured with QLSI (left) and HD-QLSI (right) techniques</figcaption></figure>

On QLSI phase images at low magnification (10x), we can see that there are reconstruction problems occurring at the bead edges where the phase gradient is the highest. To overcome this effect, the standard solution is to increase the phase gradient sampling by increasing the magnification. Here, with the HD-QLSI, because of the higher dynamic range, there are no reconstruction problems even at 10x magnification. Theoretically, the dynamics of HD-QLSI measurements is 16 times larger than the standard QLSI.

## Acquisition rate

The main limitation of this technique is that it is not single shot and has a reduced acquisition rate with respect to standard QLSI. At least 9 interferograms corresponding to different displacements are needed to reconstruct a phase image using HD-QLSI. For instance, with a 40-Hz Zyla 5.5 camera, we acquire phase images at a rate of 4.4 Hz. In the context of living cell imaging, this is enough for population observation but not for fast moving particle tracking. For the latter, it is necessary to reduce the camera field. However, the

acquisition rate is limited by the grating translation stage speed which in our prototype case was set to 100 Hz.

Gradient field recovery is just a matter of linear algebra and matrix multiplication. This is easily parallelized, and the algorithm could efficiently be implemented on a Graphical Computer Unit (GPU). We found out that the phase recovery process is then around 7 faster with GPU than with CPU. At full camera field (5.5 Mpixels), the processing time was 960 ms.

## Imaging of biological samples

This new technique can be used for any QPI applications but also in optical metrology, laser characterization. In this paper, we will focus on microscopic QPI for biological applications. Indeed, as described before, the prototype introduced in section 3 can be easily plugged onto a microscope.

### Large field of view quantitative phase images with subcellular resolution

We first demonstrated the ability of HD-QLSI to acquire large fields of view containing thousands of cells as well as image intracellular cell components with high contrast. For this purpose COS-7 cells were incubated in DMEM supplemented with 10% foetal bovin serum and 1% antibiotics (Fisher). They were placed in a glass bottom µdish (Ibidi, Martinsried, Germany) coated with poly-L-lysine.

At low magnification (10x) (see Figure 7), the large field and the measurements high dynamic enable to image thousands of cells in a single image. The high phase resolution and artifact-free HD measurements enable to segment these cells and extract relevant biological features such as dry mass, average density or other geometrical features and follow parameters over time as this technique is noninvasive and not phototoxic. This technique is particularly relevant to process high content screening as we can record information on a large cell population with a single image.

Moreover, if we zoom in, we can easily distinguish cells in G0/G1/S and early G2 states when cells are well spread on the coverslip, whereas cells close to mitosis or in mitosis are round and denser. Therefore, from density and shape features, we can easily assess the ratio of cells under mitosis over the total number of cells, which is known as the mitotic index. This is a frequently used indicator for tumor invasiveness or grade.
<figure><img src="image_8.png" /></figure>

Figure 7: (1) Quantitative phase image of fixed HeLa cells (5x; NA 0.15) (2) Zoom inside image (1). The variations in the background are due to inhomogeneities of the fixing and mounting chemicals. These are already partly compensated by subtracting a 4th order polynomial fit on the background.

## High contrast full cell imaging with organelle and cytoskeleton resolution

At high magnification, HD-QLSI reveals the cell internal components with a quite high contrast without the need for labelling. HD resolution enables to resolve intracellular vesicles, even close to the nucleus while the entire cell is imaged (see Figure 8). This is not possible with standard QLSI, though previous studies were made on partial images of the cell [30] [31]. We can detect a lot of vesicles (endosomes, lysosomes, peroxisomes) which size ranges from 40 to 3000 nm, below and above the diffraction limit, depending on their OPD. Mitochondria (1-6 µm) are also clearly detectable. This paves the way to intracellular tracking of organelles or mitochondria.

Interestingly, if a band-pass filter is applied to a QPI image at high magnification, part of the cytoskeleton is revealed, mostly microtubules (see Figure 9). For the cytoskeleton we can detect actin fibers (bundles of several individual microfilaments of 8 nm in diameter) and microtubules (25 nm of external diameter) [32]. Since the OPD of such structures is close to 1 nm, averaging is necessary to bring them above the detection level. Therefore, timelapse videos could study organelle trafficking along this dense network and how its disorganization alters the cell behavior.

<figure><img src="image_9.png" /></figure>

Figure 8: HD phase image of a living COS-7 cells at 100x magnification NA 1.3

<figure><img src="image_10.png" /></figure>

Figure 9: HD Quantitative Phase images of COS-7 cells at 150x magnification NA 1.3 (10 times averaged images, high-pass filtered – 5% of Nyquist)

## Fluorescence correlative imaging

These studies in low and high magnification configurations confirmed the interest of HD measurements for biological applications. Another advantage of this technique is to combine the phase and fluorescence modalities. Since the lateral resolution of the HD intensity images is not degraded, the calculated HD intensity makes it possible to obtain conventional fluorescence images. Moreover, it is straightforward to acquire HD fluorescent and phase images with the same sensor. For this purpose, a near infrared light source (~800 nm) is used to produce phase images. This spectral range is transmitted by the dichroic mirror used to discriminate fluorescence excitation and emission. The superposition is then almost perfect and straightforward, being only limited by the microscope objective lateral chromatism. The specific information provided by fluorescence modality can then easily be combined (see Figure 10) with the morphological and quantitative information of the phase modality.
<figure><img src="image_11.png" /><figcaption>Figure 10: Fixed U2-0S cells (40x; NA 0.6) tagged with AlexaFluor 488 phalloidin labelling actin filaments. (1) fluorescence image, (2) phase modality, (3) composite correlative image.</figcaption></figure>

On these images, we can study the actin network with fluorescence. However the high cell density makes it impossible to attribute actin fibers to individual cells. QPI images makes this association possible and easy. This is even more obvious in the case of Figure 11. In this study, the Golgi apparatus is labelled for fluorescence. Quantitative Phase Imaging reveals where it is located with respect to the cell nucleus. With fluorescence-only imaging, it would be necessary to add a fluorescence labelling, which complexifies the imaging process, when the nucleus information is not at the heart of the biological question.

<figure><img src="image_12.png" /><figcaption>Figure 11: Fixed U20S cells (100x; NA 1,3) tagged with m-Cherry to identify the Golgi apparatus. (1) fluorescence image, (2) phase modality, (3) composite correlative image. In this case, we chose to change the LUT to gray shades to make the red color more visible.</figcaption></figure>

## Faster QPI tissue slide scanning

Imaging and scanning a tissue deposited on a microscope slide often requires a long process to dye or label specific molecules or structures. This is lengthy and requires high skills. Quantitative phase imaging only requires a simple fixation and mounting process to obtain highly contrasted images of tissues slices (see Figure 12).

What HD-QLSI brings to standard QLSI is the possibility to increase the slide scanning speed. The definition factor is closely related to the number of images necessary to scan a specific area. In other words, if we increase the definition factor by a factor 4, we will reduce the number of acquired images by a factor 16, while keeping the same spatial resolution at the sample level. Of course, we need 9 acquisitions to generate a HD-QLSI image, so that the acquisition time for the same area is only 16/9 faster. However, with standard QLSI, more moves are necessary and stage translation is a very slow process compared to HD-QLSI acquisition. Consequently, the gain in slide scanning time is expected to be 16-fold.

<figure><img src="image_13.png" /><figcaption>Figure 12: Quantitative phase image of mouse skin tissue (15x; NA:0,3) on a large field of 1.1 mm x 0.9 mm. Thickness: 3 µm, fixation: formaldehyde, mounting medium: PERTEX</figcaption></figure>

## Discussion and conclusions

In this paper we presented a new High-Definition Quadwave Lateral Shearing Interferometry (HD-QLSI) technique which produces phase and intensity images with the same number of pixels as the original camera images. We presented the general principle of High Definition for any kind of Multiwave Lateral Shearing Interferometers and in particular for QLSI. We first validated our model with simulations. We then created a working prototype and validated its performance with quantification, resolution, dynamic range and acquisition rate measurements. We finally illustrated this technique with possible biological applications.

The main limitation of HD-QLSI, compared to QLSI, is that it requires 9 sequential acquisitions and is thus not compatible with short events. This could be a problem for laser metrology or microscopy of fast moving objects. However, the temporal bandwidth currently is 4.4 Hz for the full camera field. This is enough for tissue imaging or cell growth observation. In particular, the improvement for whole slide scanning in histology is impressive since fast grating translations replace slow and sometimes unprecise stage translations.

HD-QLSI opens the way to low magnification quantitative phase imaging and high content screening. More information is acquired in a single image and whole slide scanning is possible within minutes. Finally, perfectly registered phase-fluorescence images are recorded with state-of-the-art fluorescent imaging.

Other possible applications can be metrological measurements where a large dynamic range is needed. Large materials areas are also imaged in single shot. With this improvement, QLSI now combines the versatility and dynamics of a wave front sensor with the resolution and precision of a Fizeau interferometer.

## Fundings
This material is based upon work partly supported by the ANRT (Association nationale de la recherche et de la technologie) with a CIFRE fellowship granted to Roman ZINCHUK in collaboration with Institut Fresnel.

## Disclosures

All authors affiliated with PHASICS are employees or former employees of the private company PHASICS SA, France. Besides the authors declare that there are no conflicts of interest related to this article.

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