Quantitative phase microscopy using quadriwave lateral shearing interferometry (QLSI): principle, terminology, algorithm and grating shadow description
QQuantitative phase microscopy using quadriwavelateral shearing interferometry (QLSI): principle,terminology, algorithm and grating shadowdescription.
Guillaume Baffou
Institut Fresnel, CNRS, Aix-Marseille University, Centrale Marseille, Marseille, [email protected]
ABSTRACT
Quadriwave lateral shearing interferometry (QLSI) is a quantitative phase imaging technique based on the use of a diffractiongrating placed in front of a camera. This grating creates a wire-mesh-like image, called an interferogram, that is postprocessedto retrieve both the intensity and phase profiles of an incoming light beam. Invented in the 90s, QLSI has been used innumerous applications, e.g., laser beam characterization, lens metrology, topography measurements, adaptive optics, orgas jet metrology. More recently, the technique has been implemented on optical microscopes to characterize micro andnano-objects for bioimaging and nanophotonics applications. However, not much effort has been placed on disseminatingthis powerful technology so far, while it is yet a particularly simple technique. In this article, we intend to popularize thistechnique by describing all its facets in the framework of optical microscopy, namely the working principle, its implementationon a microscope and the theory of image formation, using simple pictures. Also, we provide and comment an algorithmof interferogram processing, written in Matlab. Then, following the new extension of the technique for microscopy andnanophotonics applications, and the deviation from what the technique was initially invented for, we propose to revisit thedescription of the technique, in particular by discussing the terminology, insisting more on a grating-shadow descriptionrather than a quadriwave process, and proposing an alternative appellation, namely "grating shadow phase microscopy" or"grating-assisted phase imaging".
The wave nature of light gives a special importance to the concept of phase in optics. In the scalar approximation, a static,monochromatic light field can be represented by a complex field E ( r ) = E ( r ) exp ( i ϕ ( r )) , where E ( r ) ∈ R is the electric fieldamplitude and ϕ ( r ) ∈ R is its phase, at the position r . When probing a light field, the easily accessible physical quantity isthe intensity I ( r ) ∝ | E ( r ) | . But this quantity only provides a partial information. Accessing also the phase of a light field ispossible but less direct, and usually requires more sophisticated techniques involving interferences, as a means to convert thephase information into measurable intensity modulations. The techniques capable of mapping the phase of a light field arecoined quantitative phase imaging (QPI) techniques.This article focuses on one particular QPI, that is named quadriwave lateral shearing interferometry (QLSI). This techniqueis far from being the most popular, although it gathers many advantages. This article is aimed to favor the dissemination of thetechnique. We first describe its principle. Then, we discuss the terminology used in QLSI introduced 20 years ago, and proposeto revisit it, in order to make it simpler, more understandable and consistent, in particular for the rising applications in opticalmicroscopy and nanophotonics. Finally, we provide and describe a Matlab code, 20-line long, suited to retrieve the intensityand phase images from a raw image camera. Quadriwave lateral shearing interferometry (QLSI) is a quantitative phase imaging (QPI) technique, capable of measuring boththe intensity and the phase of a light beam, with high spatial resolution, and high sensitivity, in a particularly simple manner.This technology, imagined in the 90s by J. Primot, and patented in 2000, has been mainly utilized by the community usingcommercialized devices sold by only one company worldwide so far (Phasics S.A.). Originally designed for precise laser beamcharacterization , the scope of applications extended in the 2000s to lens metrology, X-ray imaging, adaptive optics and a r X i v : . [ phy s i c s . op ti c s ] F e b urface characterization. In 2009, QLSI was proven powerful when used in an optical microscope, widening the range ofQLSI applications to the study of small objects in biology and very recently in nanophotonics.
Wavefront analyser Chessboard grating (CBG)CBG camera bInterferogramd noise gTransmittancee f Optical thickness
Living cell Temperature gradient MoS layer Gold nanoparticle Metasurface ha (nm) (nm)OT (nm) -151 23
10 µm
Transmittance µ m µ m µ m
10 µm c refractive object planarwavefrontdistortedwavefront = 0.039 nm Figure 1. (a) Schematic of a QLSI wavefront analyzer, composed of a regular camera equipped with a chessboard diffractiongrating. (b) Representation of a typical QLSI chessboard diffraction grating. (c) Schematic showing the wavefront distortion δ ( x , y ) experienced by a collimated light beam due to the presence of a metasurface. (d) Example of a raw QLSI image, calledan interferogram, corresponding to a 2-µm dielectric bead. (d,e) Intensity and optical thickness images retrieved from theinterferogram (d). (g) Image of the intrinsic noise in an OT image, with 3 s exposure time, characterized by a standarddeviation of σ = .
039 nm (Zyla camera (Andor) and Sid4Element reimaging system (Phasics)). (h) Examples transmittanceand OT images of micro- and nano-objects acquired by QLSI, namely a living cell, a wavefront distortion created by a localmicrometric induced temperature temperature in water, a molybdenum disulfide flake, and gold nanoparticle and ametasurface. QLSI is based on the use of a so-called wavefront analyzer that consists of the association of two simple elements: a regularcamera and a 2-dimensional (2D) diffraction grating, separated by a couple of millimers from each other (Fig. 1a). Thegrating affects both the phase and the intensity of light: Opaque horizontal and vertical lines are blocking the light, and definingtransmitting square areas imprinting phase shifts of 0 and π on the transmitted light, according to a chessboard pattern (Fig.1b). Upon illumination, the diffraction grating creates an image, called an interferogram (Fig. 1d), on the camera sensor thatcan be processed to retrieve both the intensity (Fig. 1e) and the wavefront profile W ( x , y ) (Fig. 1f) of a light beam . When thewavefront distortion W originates from an imaged object (Fig. 1c), the mapping of W can be used to optically characterize the bject. In this case, instead of wavefront profile, one rather speaks about the optical path difference (OPD), or equivalently theoptical thickness (OT) of the object δ (cid:96) = W , defined as δ ( x , y ) = ( n − n ) h ( x , y ) , (1)where n is the refractive index of the object and n the refractive index of the surrounding medium (Fig. 1c). When the object isnot uniform, this more general expression applies: δ ( x , y ) = (cid:90) h ( x , y ) ( n ( x , y , z ) − n ) d z . (2)One also sometimes refers to the phase of the light ϕ ( x , y ) , which can be calculated from W provided the illuminationwavelength λ is known: ϕ = πλ W (3)Thus, calling QLSI a phase imaging technique is somehow inconsistent, as it primarily measures a wavefront profile. Strictlyspeaking, it is more a wavefront sensing technique than a QPI. Retrieving the phase requires a preknowledge (the wavelength),and the phase may not be accurately defined in the case of a broadband illumination. All these physical quantities, phase, OPD,OT and wavefront profile, are equivalent, interchangeable, and all used in the literature.The advantages of QLSI over other existing QPI techniques are many-fold. (i) An interesting advantage is the achromaticity.Counterintuitively, although based on the use of a grating, not only the technique can be used with broadband illumination, butalso the knowledge of the illumination wavelengths is not necessary to calculate the intensity and wavefront images from theinterferogram (the reason is explained later on). This feature makes QLSI particularly robust. (ii) The noise standard deviationin OPD measurements is typically 0.6 Å · Hz − / , i.e., ∼ − λ · Hz − / corresponding to around 1 mrad · Hz − / of phase delayin the visible range. (iii) It provides an evaluation of the error from the measurement itself, a unique concept in QPI. (iv) Thespatial (i.e., lateral) resolution reaches the diffraction limit, unlike other popular wavefront sensors such as Shack-Hartmann.(v) It is easy to use, as it just consists in using a camera-like device. No modification of the microscope is required. (vi) QLSIbenefits from the high sensitivity of interferometric methods but do not suffer from their usual drawbacks: it neither requiresa reference beam, nor a complex alignment that might be sensitive to external perturbations. The relative positioning of thegrating with respect to the camera is done once and for all, and is not sensitive to, e.g., temperature variation, mechanical driftor air flow.Albeit conceived in the 90s, the idea of plugging a QLSI device on a microscope is recent. Figure 1h draws an overview ofthe main applications of QLSI in optical microscopy, namely cell imaging, temperature imaging in nanoplasmonics, single nanoparticle optical characterization and metasurface characterization (Figure 1h).QLSI has the ability to become one day a technique of predilection for quantitative phase imaging applications. One canregret it is not already the case after more than 20 years. The current cost of commercial QLSI devices and the long-standingmyth that QLSI is sophisticated presumably explains the reluctance of the community to buy it, or to set it up. Also thecomplexity of the name "quadriwave lateral shearing interferometry" may be reluctant and help perpetuate the myth of anunaccessible technique. The aim of this article is to favor the dissemination of the technique and contribute to provide QLSIwith the attention it deserves, in particular by breaking this myth. All the magic of QLSI happens between the grating and the camera, over a couple of millimeters. What is happening in therecan be understood in two ways. The first picture deals with the interference between four diffraction orders of the grating,hence the name of the technique. The second picture, less popular albeit simpler, refers to a shadowing effect of the grating,very similar to the working principle of a Shack-Hartman wavefront sensor. Both of them are as important. Depending on thecommunity or the application under consideration, one picture could be more appropriate than the other. In this section, focusin put on the understanding of the working principle of QLSI, using simple pictures.
The 0 − π chessboard-pattern of the grating, depicted in Fig. 1b, is the key feature to make QLSI work. This phase shiftarrangement makes the diffraction grating diffraction-less , as strange as it may seem. Figure 2 explains this behavior byshowing calculations of the light propagation between the grating and the camera, for different grating properties, from simpleto complex. Calculations have been performed for a single unit cell of the grating (such as ), with periodic conditions.
00 2.65 µm2.65 µm80 µm monochromaticpolychromatic
80 µm000 abcdef
000 39 µm39 µm3 mm3 mm00000 39 µm39 µm3 mm3 mm0
Figure 2.
Modelling of the light propagation (from left to right) after various diffraction grating geometry. (a) Grating period: Λ = . / λ =
550 nm, no chessboard phase pattern. (b) Λ = . / λ =
550 nm, 0 − π chessboard phase pattern.(c) Λ =
39 µm, λ =
550 nm, no chessboard phase. (d) Λ =
39 µm, λ =
550 nm, 0 − π chessboard phase pattern. (e) Λ = λ = [ , ] nm, no chessboard phase. (f) Λ =
39 µm, λ = [ , ] nm, 0 − π chessboard phase pattern.Figure 2a starts with a grating without 0 − π alternation (unit cell: ). The transmitted light features complex and contrastedinterferences, as expected for a grating. A periodicity of the pattern in the z direction can be observed, a property coming fromthe Talbot effect, stating that the grating is re-imaged at periodic distances separated by Z T = Λ / λ . (4)where Λ is the grating’ period. Interestingly, when the 0 − π phase shift alternation is restored (Fig. 2b), then theinterferences fully disappear, and the transmitted light becomes invariant by translation along ( Oz ) , propagating like a normalshadow behind an opaque object. This peculiar behavior comes from the cancellation of the zero diffraction order by forwarddestructive interferences, so that only the four 1st orders of the transmitted light remain. The transmitted light pattern isthus equivalent to the interferences observed after a Fresnel biprism, in 2 dimensions. These two introductory cases (Figs.2a,b) do not exactly match commonly used QLSI gratings, since the grating periodicity in these calculations was very small( Λ = .
325 µm). QLSI wavefront sensors rather feature a periodicity corresponding to 6 to 8 times the periodicity of the camerasensor (i.e., typically 6 × . =
39 µm) in order to resolve the interferogram modulation. For Λ =
39 µm, the shadowing effectdepicted previously remains if a 0 − π chessboard pattern is applied (Fig. 2d), but the beam exhibits some inner structuration.Interestingly, these structurations disappear once a polychromatic beam is used. Each wavelength creates its own structuration,and they all cancel each other out after a given distance that reads Z P = (cid:114) − ln ( V ) π Λ ∆ λ , (5)where ∆ λ is the spectral width of the illumination and V the tolerated blurring of the interference pattern (0 < V < V = / e , one gets the expression: Z = − √ π Λ ∆ λ (6)This effect, depicted by Primot et al. in 2000, was coined the panchromatic Talbot effect, and Z P the panchromaticdistance. This shadowing effect and invariance by z -translation makes the grating-camera distance not so critical. A proper,well-defined array of dots will be obtained in any case, provided the grating is placed around Z P or above, as shown in Fig. 3.Figure 3 gives an overview of the general working principle of QLSI. The size of the diffraction grating has been reduced to6 × =
36 unit cells for the sake of clarity. In practice, it is rather composed of more than 300 ×
300 unit cells, to cover the ize of the camera sensor. The camera (here positioned at 1 mm from the camera) collects a shadow-like image of the grating,composed of periodic dots arising from the interferences between four diffraction orders, two along the x direction and twoothers along the y direction, the zero order being cancelled by destructive interferences created by the 0 − π chessboard pattern.Also, the 2 / This splitting infour direction of propagation creates 4 images on the camera sensor, only slightly shifted with each other typically by an angleof θ = . ◦ , following Bragg’s law, 2 Λ sin θ = λ , where Λ =
39 µm is the grating period. ++ Figure 3.
Overview of the working principle of QLSI. The size of the diffraction grating has been reduced to 6 × × The shadowing picture depicted above provides a simple explanation of the working principle of the technique, but haslimitations. Figure 5 shows what happens when one hole of the grating is blocked. As expected, no light is transmitted outsidethis hole, following the shadow picture, but after a few 100s of microns, the light beam is recovered. This effect is usually called"beam self-healing". Thus, this simple shadow picture should be used with caution. It can be used to popularize the principlemaking it understandable by the layman, simply explain why the technique is achromatic (see next section), etc, but not toproperly investigate the underlying physics and advance science. The light propagation after the grating remains a diffractionprocess and to carry out fundamental research in grating-assisted phase microscopy, the multiwave picture has to be considered.
Interestingly, the shifts of the dots represented in Fig. 4 do not depend on the wavelength, only on the wavefront gradient, as itmimics a simple shadowing effects. This is the reason why QLSI is achromatic, although it is based on a diffraction grating. Note that some algorithms reduce the number of pixels in the phase image by a factor of 9 or 16 to accelerate image processing. a w a v e f r o n t b w a v e f r o n t cd µ m
100 µm w a v e f r o n t w a v e f r o n t Figure 4.
The QLSI principle understood as a shadowing effect. (a) Calculation of the transmitted light intensity after a 0 − π chessboard grating, for a planar incoming light wavefront at normal incidence. For the sake of simplicity, the square gratingcontains 4 × ◦ , producing a lateral shift of the shadow according to this same angle. (c) Simulation of transmitted light whenthe wavefront is convexe, producing a spreading of the shadow. (d) Simulation of transmitted light when the wavefront isconcave, producing a shrinking of the shadow. w a v e f r o n t Figure 5.
Calculation of the transmitted light intensity after a 0 − π chessboard grating, for a planar incoming light wavefrontat normal incidence, where one hole of the grating is blocked. A self-healing of the beam is evidenced, showing the limitationof the shadowing picture.The knowledge of the wavelength is not even necessary to reconstruct the wavefront image from the interferogram. Only thegrating-camera distance matters.However, it does not mean that the wavelength has no effect on the interferogram quality, and that a given QLSI wavefrontsensor can be used for any wavelength without problem. To imprint the π phase shifts in the chessboard grating, the transparentsubstrate is locally etched to remove a thickness h such that π = πλ ( n s − ) h . For glass, one has n s ≈ .
5, which gives h = λ .Thus, a diffraction grating is made for a specific wavelength a priori. If used for another wavelength, the phase shifts deviatefrom π . The principle depicted in Fig. 4 still applies, but the contrast of the interferogram will be reduced. In other words, themeasures are still quantitative, but the signal to noise ratio is poorer and the sensitivity reduced. However, this limitation is notdramatic. Typical QLSI wavefront sensors suited for visible wavelengths can be used in the 450-750 nm range without much roblem. In the Fourier space discussed in the next section, an unadapted wavelength produces additional diffraction spots, thatcan still be removed numerically, if need be. vs GS How come a simple grating in front of a camera can be coined quadriwave lateral shearing interferometry? This questionis often raised by the layman and the answer is not trivial, as detailed in the previous section. Our experience is that thisterminology may even be reluctant and give a negative prejudice regarding the complexity of the method, especially for the newcommunities starting using this QPI technique (namely optical (bio)microscopy and nanophotonics): the complexity of thename "quadriwave lateral shearing interferometry" does not reflect the simplicity of the technique, experimentally speaking. Thereason of the name QLSI is historical: The technique resembles a previous QPI technique named lateral shearing interferometry(LSI), based on the interferences of two light beams, tilted with each other using a prism. But LSI does not involve a gratingand is very different experimentally speaking from QLSI. We believe that, at some point, one should detach from historicalconsiderations and focus on scientific considerations, to reach a better description. Other, descriptions are made at the expenseof the simplicity and clarity.This chessboard grating is the central (and even the only) part of the QLSI technique. The grating is what makes thistechnique unique, different from all the other phase imaging and wavefront sensing techniques. For this reason, the use of theword "grating" within the name of the technique would be justified and natural. "Quadriwave lateral shearing interferometry"neither tells what the technique is (a grating in front of a camera), nor what it does (wavefront sensing or a phase imaging).For these reasons, names such as "Grating shadow (GS) phase microscopy", "Grating shadow (GS) wavefront sensing" or"Grating-assisted phase microscopy" would certainly ease the dissemination of the technique, its reference, its explanation, andwould contribute to break the myth of a sophisticated and out-of-reach technique for the layman. Alternatively. Changing thename of a 20 year-old scientific field, device, technique or concept is difficult, albeit not impossible. Many examples exist. Itusually takes time and gives rise to some reluctance from the initial community, following the famous Planck’s principle. vs mask Until here, in this article, we have been using the name "grating" to define the diffraction element, which seems natural. Yet,the community usually prefers the name "modified Hartmann mask".
We also propose here to revisit this appellation tofavor a better understanding. First, the grating has little to do with a Hartmann mask. Originally, a Hartman mask is not even agrating. A Hartman mask consists of an opaque screen pierced by several holes (originally three, sometimes more), to helpadjust the focus of a telescope. Second, the main aim of the chessboard grating is not to mask part of the incoming beam. Onthe contrary, it is to engineer the transmitted light. Thus, it makes more sense to call it a grating rather than a mask. Again, thereason of the "mask" appellation is more historical than scientific, making it difficult to understand.Born & Wolf, in their seminal book, defined a grating as "any arrangement which imposes on an incident wave a periodicvariation of amplitude or phase, or both", which is exactly what the chessboard grating does. Interestingly, they also notedthat the common analysis of a one-dimensional grating "may easily be extended to two- and three-dimensional periodicarrangements of diffracting bodies", but they stated that, unlike 1D or 3D gratings, "2D gratings (called cross-gratings) foundno practical applications". It seems time has changed!Interestingly, the use of a grating is not even necessary. It certainly contributes to optimize the interferogram quality, butnon-periodic, and even random, optical elements have been shown efficient. A particularly simple and cheap approach consistsin replacing the grating with a thin diffuser, creating a speckle pattern on the camera sensor, instead of the well-defined, periodicarrangement of dots. Then, the principle just consists in monitoring the distortion of the speckle pattern to retrieve the wavefrontprofile. This technique has been recently pioneered for optical microscopy developments, but was proposed long ago formore general applications. vs Bessel
The translational invariance of the transmitted light after a chessboard grating (provided a polychromatic light is used, see Fig.2f) has been coined the "panchromatic Talbot effect". However, a Talbot effect is rather a phenomenon of image replication atperiodic distances, not an invariance by translation. What happen between the grating and the sensor rather belong to the familyof propagation-invariant optical waves, or non-diffracting beams, the most famous members of this family being the Besselbeams. The nature of the optical wave could thus be referred as a Bessel-like wave, produced when a polychromatic light isused. A similar non-diffracting beam shape was also called a Cosine-Gauss beam. Once again, the reason of the reference toa Talbot effect is historical, because it has been discovered upon studying the Talbot effect.
Image retrieval algorithm
DFT -1 DFT rotation.arg(DFT -1 ) InterferogramFourier space 1 st order along x’ Integration phase gradient along y’phase gradient along x’1 st order along y’0 th orderintensity Phase x’y’ D e m o d u l a t i o n .arg(DFT -1 ) phase gradientalong phase gradientalong Figure 6.
Summary schematic that displays the whole intensity and phase retrieval procedure at once.The section is aimed to show that numerically retrieving the phase of light in GS microscopy is simple. A description of thisprocedure can be found in Refs. (in French). The image displayed in Figure 1c is called the interferogram. It is the raw dataacquired by the camera sensor, and resembles the intensity image (Figure 1c), on top of which the shadow of the grating looksprinted. When the incoming wavefront is perfectly planar, the fringes of the grating’s shadow are perfectly parallel. Whenthe incoming wavefront is non uniform, an imperceptible distortion of these fringes appears, which creates deviations froma perfect spatial periodicity that can be extracted by a demodulation algorithm involving Fourier analysis. This is the basicprinciple of the retrieval algorithm that we shall explain in this section. The overall numerical procedure is schematized in igure 6. Let us break it down and explain it step by step, and illustrate it using Matlab codes.Let I be the interferogram image, say a N x × N y matrix:The overall schematic of the algorithm for intensity and phase images retrieval is given at the end of this article, in Figure 6. Let ˆ I be the Fourier transform of I :ˆ I = F [ I ] (7) Matlab code> I=imread(’interferogram.tif’);> Ihat=fftshift(fft2(I));
DFT
Interferogram Fourier space
Figure 7. (right) Raw interferogram I acquired by the camera and (left) representation of its Fourier transform ˆ I .Figure 7 displays the discrete Fourier transform (DFT) of the interferogram. In addition to the central spot, four main peripheralFourier spots are visible. They correspond to the fringes of the interferogram, and contain the information on the phase of thelight beam. The positions of these spots define two directions ( x (cid:48) ) and ( y (cid:48) ) . These directions are tilted by an angle θ comparedwith the original ( Oxy ) frame because the grating is tilted itself by this angle. In this image, the distance between the centralspot and a peripheral spot is 1 / Λ /
2, half the grating period)equals three times the pixel size. This configuration optimizes spatial resolution of the processed phase and intensity images.
The second step consists in isolating the 1st order diffraction spots. This is where the information related to the phase iscontained. There are 4 spots, i.e., 4 options a priori. However, diametrically opposed spots are redundant. They contain exactlythe same information since they are exact complex conjugate. This symmetry in the Fourier space comes from the fact that theoriginal image I is real. Thus, only two spots can be considered, without loss of information, any of them provided they are 90 ◦ apart.For both spots, the process is the same. The spot is cropped by a disc of radius R . This disc should be small enough, toavoid gathering information from neighboring spots, but not too small to avoid too much removal of high spatial frequencies ofthe image, and a blurring of the final phase image. R typically equals one sixth of the image. If the image is not square, then thecropped area is an ellipse. Then, the cropped spot is centered in the Fourier space (Figure 8). > %% Demodulation of a diffraction spot > % Crop of the diffraction spot> [Ny,Nx] = size(Ihat);> [xx,yy] = meshgrid(1:Nx,1:Ny);> dx = 127; dy = 210; % position of the first order spot to be cropped> R2C = (xx-Nx/2-1-dx). ˆ ˆ ˆ ˆ ourier space1 st order along y’ 1 st order along x’ croprecentering recenteringcrop Figure 8.
Schematic of the demodulation procedure in the Fourier space. > Ihatc = Ihat.*circle;> % Recentering of the cropped spot> H1=circshift(Ihatc,[-dy -dx]); % recentering
This procedure is to be repeated for the second spot, leading to two images that we name H and H , correspondingrespectively to the spots along x (cid:48) and y (cid:48) . The images H and H shall be inverse-Fourier transformed. Back in the ( x , y ) space, the fringes are now gone, since the spotwas centered in the Fourier space. However, The recovered image has no reason to be real, since the crop and translation in theFourier space cancelled the Hermitian property of the Fourier image. The values of the recovered images are thus complex and,interestingly, the argument of these complex values is proportional to the local phase gradient ∇Φ ( i , j ) . The proportionalityconstant depends on the periodicity Λ of the chessboard grating, and on its distance d to the sensor: ∇ x (cid:48) Φ = Λ √ π d arg (cid:0) F − [ H ] (cid:1) , (8) ∇ y (cid:48) Φ = Λ √ π d arg (cid:0) F − [ H ] (cid:1) . (9)In practice, the factor is not calculated this way. The factor is measured experimentally, using a reference sample of knownoptical thickness (e.g., grooves on a glass sample characterized by AFM). Before integrating the vectorial gradient to get thephase, one needs to rotate it by an angle − θ to retrieve gradients over the axes ( x ) and ( y ) : (cid:20) ∇ x Φ∇ y Φ (cid:21) = (cid:20) cos ( θ ) − sin ( θ ) sin ( θ ) cos ( θ ) (cid:21) · (cid:20) ∇ x (cid:48) Φ∇ y (cid:48) Φ (cid:21) (10) > %% Inverse Fourier transform > Ix = ifft2(ifftshift(H1));> Iy = ifft2(ifftshift(H2));> factor = 0.52;> %% Phase gradient calculation > DPh1 = factor*angle(Ix);> DPh2 = factor*angle(Iy); %% Phase gradient rotation > DPhx = cos(theta)*DPh1-sin(theta)*DPh2;> DPhy = sin(theta)*DPh1+cos(theta)*DPh2; This part of the algorithm is depicted in Figure 9. rotation.arg(DFT -1 ) st order along x’ phase gradientalong xphase gradientalong yphase gradientalong y’phase gradientalong x’1 st order along y’ .arg(DFT -1 ) Figure 9.
Schematic of the retrieval of the phase gradients from the Fourier images H and H . The final step consists in a very fundamental mathematical task: retrieving a 2D scalar field from its vectorial gradient. Severalalgorithms are freely availabe, some of them written in Matlab. For instance, the Matlab package from John D’Errico does agood job. > %% Phase gradient integration using the John D’Errico algorithm > Pha = intgrad2(DPhax,DPhay); Usually here, an artificial phase tilt may occur throughout the image. This tilt can be removed by removing the Zernikemoment Z of the image. A Matlab package for Zernike moment analysis was developed by Amir Tahmasbi. > %% Removal of the global phase tilt > Z = ZernikeMoment(Pha,1,1);> r0=min(Nx,Ny)/2-1; x0=Nx/2; y0=Ny/2;> R = sqrt((2.*xx-2*x0-1).ˆ2+(2.*yy-2*y0-1).ˆ2)/(2*(r0+1));> Theta = atan2((2*y0-1-2.*yy+2),(2.*xx-2*x0+1-2));> Rad = radialpoly(R,n,m);> Zimage=0.5*Rad.*(Z’*exp(-1i*m*Theta)+Z*exp(+1i*m*Theta));> Pha=Pha-Zimage; Not only the phase can be retrieved from the interferogram, the intensity image can also be calculated using a similar procedure.This time, the central spot (zero order) has to be cropped in the Fourier space, by the same disc of radius R . ntegration Phasephase gradientalong phase gradientalong
Figure 10.
Schematic of the integration of the phase gradient to retrieve the phase image. > % Crop of the diffraction spot> R2C = (xx-Nx/2-1). ˆ ˆ ˆ ˆ DFT -1 Fourier space 0 th order intensity crop Figure 11.
Schematic of the intensity image retrieval alrogithm.
The way it is presented above, the procedure is slightly simplified. Used this way, the procedure would give a phase imagetarnished by a lot of imperfections, coming from microscope aberrations, pieces of dust on the optics, light beam imperfection.All these static wavefront imperfections can be, and must be, removed by acquiring another interferogram image, usually calledthe reference. In microscopy experiments, the reference is usually acquired on a blank field of view, free from any object ofinterest. If no blank area is present in the sample (like in the case of cultured cells in confluence for instance), then the referenceimage can be acquired upon moving the sample holder rapidly and randomly during the exposure time. The whole algorithmdetailed above must be also applied to the reference to retrieve the intensity reference T ref and, more importantly, the phasereference Φ ref . T ref can be used to calculate the transmittance image t = T / T ref . Φ ref must be calculated and subtracted to thephase image retrieved from the above-detailed algorithm in order to extract a proper phase profile Φ .In practice Φ and Φ ref are not subtracted at the end of the code. The subtraction can be done at the moment when the phasegradients along ( Ox (cid:48) ) and ( Oy (cid:48) ) are calculated: > %% Phase gradient calculation > DPh1 = factor*angle(Ix.*conj(Ix_ref));> DPh2 = factor*angle(Iy.*conj(Iy_ref)); Summary
In light of recent applications in optical microscopy, in particular for biomiaging and nanophotonics, we propose here to revisitthe field quadriwave lateral shearing interferometry, with simple working principle descriptions and some new terminology. Inparticular, we propose to use grating-shadow phase microscopy as a well-suited name of the technique, which should favor itsdissemination, ease its description, and contribute to make it more popular and accessible for these fields of applications. Weexplained the working principle of GS phase microscopy and highlight the main advances when implemented on an opticalmicroscope, in particular in bioimaging and nanophotonics for nanoparticle, 2D material and metasurface characterization. Wego more into detail about the working principle by describing what occurs between the grating and the camera, the only twoelements involved in GS phase microscopy. In particular, we explain that two complementary visions can be considered: a4-image interference picture, and a grating-shadow picture. Finally, we detail the image retrieval algorithm, and provide relatedMatlab codes.
Acknowledgments
The author wishes to thank D. Andrén and M. A. Alonso for helpful discussions.
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