3D mapping of refractive index exploiting Brillouin scattering

$Page 1: 3D mapping of refractive index exploiting Brillouin scattering$
Dipartimento Interateneo di Fisica

Corso di Laurea in Fisica Magistrale - Curriculum Fisica della Materia

3D mapping of refractive index exploiting Brillouin scattering

A new tool for biological samples analysis

Anno accademico 2016/2017

Relatori:

Prof.ssa Milena D’Angelo Prof. Giuliano Scarcelli

Laureando:

Carlo Bevilacqua

Contents

Introduction iv

1 Refractive index in the biomedical field 11.1 Significance and applications of refractive index . . . . . . . . . . . . 1

1.1.1 Cell biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Hematology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Pathology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 Tissue imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Techniques for measuring refractive index . . . . . . . . . . . . . . . . 4

2 Digital Holographic Microscopy for refractive index measurement 92.1 Principle of holography . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Overview on holographic configurations . . . . . . . . . . . . . . . . . 112.3 Image plane and off-axis DHM configuration . . . . . . . . . . . . . . 162.4 Phase unwrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 One-dimensional phase unwrapping . . . . . . . . . . . . . . . 192.4.2 Two-dimensional phase unwrapping . . . . . . . . . . . . . . . 212.4.3 Optical phase unwrapping . . . . . . . . . . . . . . . . . . . . 22

2.5 Refractive index measurement . . . . . . . . . . . . . . . . . . . . . . 23

3 Experimental setup for Digital Holographic Microscopy 253.1 DHM setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 Phase accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.1 Data acquisition and analysis . . . . . . . . . . . . . . . . . . 313.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Confocal Brillouin microscopy for refractive index measurement 354.1 Brillouin scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Technological challenges and solutions . . . . . . . . . . . . . . . . . 36

4.2.1 Multi-stage VIPA spectrometer . . . . . . . . . . . . . . . . . 37

CONTENTS iii

4.2.2 EM-CCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Refractive index measurement . . . . . . . . . . . . . . . . . . . . . . 41

5 Experimental setup for Confocal Brillouin Microscopy 445.1 Brillouin setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1.1 Pump laser path . . . . . . . . . . . . . . . . . . . . . . . . . 445.1.2 Signal collection . . . . . . . . . . . . . . . . . . . . . . . . . . 465.1.3 Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Spectrometer calibration . . . . . . . . . . . . . . . . . . . . . . . . . 505.3 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3.1 Data acquisition and analysis . . . . . . . . . . . . . . . . . . 535.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Conclusions and future work 59

A Branches cut unwrapping algorithms 62A.1 Phase residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A.2 Goldstein’s branch cut algorithm . . . . . . . . . . . . . . . . . . . . 63

B Spectrometer calibration 65

Introduction

As is well known, refractive index is a physical quantity defined as the ratio betweenthe speed of light in a medium with respect to vacuum. For its measurement thereare basically two approaches. The first class relies on the determination, by meansof interferometric techniques, of the optical path length inside the medium, which isjust the geometrical path length multiplied by the refractive index. The second classstudies reflection or refraction at the interface between two media having differentrefractive index; properties like reflectance or change in direction of the light arerelated to refractive indices of the two media, thus their measurement allows thedetermination of refractive index. Although there is extensive literature on theseapproaches and many variants, their extension to 3D resolved measurement is stilla challenging task. The problem is that the first approach measures the integral ofrefractive index over the whole length of the sample while the second approach worksonly at interfaces. Things are even worse for biological samples that require highresolution for measuring the small organelles inside the cells and that, due to theirmostly aqueous composition, exhibit only small variations of refractive index around1.33. Apart from its importance in basic research, a technique for retrieving a highresolution 3D map of refractive index is of great interest in biology. In fact refractiveindex has been demonstrated to measure protein contents inside cells (i.e. their drymass) and this can be useful not only for figuring out the composition of biologicaltissues but also for understanding the progress of some diseases and allow for earlydiagnosis.In this thesis, we will first examine and employ a technique that has been successfullyused over the past years for determining refractive index of biological samples. Thena new technique will be proposed and implemented, that could overcome some of thelimitations of current techniques.The first chapter will outline the importance of refractive index measurement in thebiomedical field and will briefly list the techniques that have so far been proposedand used at this end. In particular we will focus on Digital Holographic Microscopy(DHM), a technique that has gained more and more interest in last years becauseit allows quantitative phase images of a sample, from which refractive index can beinferred.The second chapter will describe the working principle of holography, the prevalent

Introduction v

configurations of DHM and, more in details, the off-axis configuration that has beenadopted for refractive index measurement in this work.In the third chapter, the focus will be on the retrieval of refractive index from phaseimages and on the characteristics of the DHM setup built at this end. Finally, exper-imental data will be presented that show how the refractive index of microspheres iscorrectly measured by this approach.Chapters four and five are specular to two previous ones. In chapter four, Confo-cal Brillouin Microscopy will be presented starting from its working principle up totechnological challenges that had to be overcome in order to make it feasible.Chapter five will focus on the novel technique that, exploiting Confocal BrillouinMicroscopy, is expected to measure the refractive index of a biological sample. Firstlythe idea behind this technique is elucidated, then the experimental setup is describedand finally a proof of principle of this measurement with liquid of known refractiveindex as a sample is given.In the conclusion of this work the measurement of refractive index by DHM and Con-focal Brillouin Microscopy will be compared. Both potentialities and disadvantagesof the novel technique will be pointed out and next steps in its development will belisted.This research work has been done during a four months internship at the bioengi-neering department, University of Maryland - College Park under the supervision ofProf. Scarcelli. The internship was financed by University of Maryland and Premiodi studio Global Thesis - Università degli studi di Bari.

Chapter 1

Refractive index in the biomedicalfield

Refractive index has always been an extremely important parameter in materialsscience since it determines the optical properties of a material. Nowadays refractiveindex is gaining an increasing interest in the biomedical field. In this chapter therelevance of refractive index measurement in the biomedical field will be pointed outand techniques that have been proposed for its measurement will be mentioned.

1.1 Significance and applications of refractive in-dex

Refractive index is an important parameter in studying cells, since it is correlatedwith their mechanical, electrical and optical properties and it quantifies protein con-centration inside the cell. Following Ref. [1], that is a recent review on the significanceof refractive index in biological studies, we present the role of refractive index in cellbiology, hematology and pathology.

1.1.1 Cell biologyThe refractive index is linearly proportional to the concentration of proteins, ncell =n0+αC where n0 is the refractive index of water or dilute salt solution, α is the specificrefraction increment and C is the mass density of protein in gram per deciliter (g/dL)[2, 3]. Thus, a measurement of refractive index can give information about proteinconcentration.Moreover, the refractive index of various organelles has been measured providing newinsights into cell biology: previously, it was widely believed that the refractive indexof the nucleus is higher than that of cytoplasm [4], but some recent studies show that

1.1 Significance and applications of refractive index 2

Figure 1.1: Applications of cell refractive index in hematology. (a) Rendered iso-surfaces of the refractive index map of a lymphocyte and a macrophage from thecross-sectional images of 3D refractive index tomograms in various axial planes. (b)Schematic illustration of the intraerythrocytic cycle of malaria infection and its 3Dphase mapping (related to refractive index) at various stages (A: healthy cell, B: ringstage, C: trophozoite stage, and D: schizont stage) with black arrows indicating themerozoite and gray arrows indicating the hemozoin. [1]

it is lower [5].Since the cell refractive index is correlated with the concentration of matter, the firstdirect application of cell refractive index measurement is cell growth monitoring [6].

1.1.2 HematologyThere are several types of cells in blood, which include red blood cells, white bloodcells and platelets; the measurements of their refractive index can be beneficial inearly diagnosis of blood diseases.

• The two types of platelets that help discriminate between healthy and unhealthyindividuals are in the form of activated and inactivated platelets, that are struc-turally very different: in the activated state, the platelets have a sphericalneedle-like structure, while the inactivated state has a discoid shape with a di-ameter of 2–4 µm and a thickness of 0.5–2 µm. Due to this difference, plateletsare a useful means to determine certain diseases such as cerebrovascular disease,

1.1 Significance and applications of refractive index 3

ischemic heart disease and renovascular disease [7, 8, 9].

• White blood cells are the body’s main line of defense. Researchers have shownthat lymphocytes from infected or vaccine injected animals have a higher re-fractive index equivalent to 1% to 2% of the protein concentration due to theproduction of antibodies [10, 11]. In figure 1.1a is shown the refractive indexmap of a lymphocyte and a macrophage.

• As human red blood cells lack a nucleus and cellular organelles, by obtainingthe refractive index of the cytoplasm of the red blood cells, the concentration ofhemoglobin can be determined, i.e. nrbc − n0 = βChemoglobin, where n0 = 1.335is the refractive index of cell fluid without hemoglobin and β = 0.0019 dL g−1

[12]. The hemoglobin concentration is related to some diseases; for example, forhypochromic anemia patients, it is about 30% lower than for healthy people.Another major disease of red blood cells is malaria with 250 million peopleinfected yearly. Hence it is crucial to detect malaria more effectively, allowingearly treatment and reducing fatality. The red blood cell refractive index andmorphology can be used as important parameters for diagnosis of diseases suchas malaria and anemia, since a significant decreasing trend in the refractiveindex of the red blood cells is shown during the infection process. In figure1.1b there is a schematic illustration of the intraerythrocytic cycle of malariainfection showing the refractive index variation during the infection process.

1.1.3 PathologyRefractive index variations have been demonstrated to be a marker for cancer cellsand infection processes.

• Various research studies have characterized the refractive index of normal andcancer cells, aiming to better understand the abnormal cell cycles and increasedproliferation of the cancer cells. Most normal cells have a refractive index of1.353 and cancer cells have a higher refractive index ranging from 1.370 to 1.400[13, 14]. As many human cancer cells show atypical cell cycles and increase incell proliferation, the increase in the refractive index may be related to theincrease in cell production during various stages of cell cycle in cancer patients.

• Prevention of parasite infection through detection of bacteria and virus in waterand food has been an ongoing process with vast improvements over the yearsdue to new technologies that reduce the amount of detection time and costs.The current standard method used for bacteria detection in drinking water isthe USEPA Method 1604 [15]. However several technological limitations existin this method that impede its effectiveness in preventing a bacteria outbreak,

1.2 Techniques for measuring refractive index 4

such as laboratory-based and long processing time. Several potential methodshave come up and one of them is using the refractive index and morphology ofdifferent bacteria to correctly identify the source of an infection [16].

1.1.4 Tissue imagingImaging of biological tissues with traditional light microscopy can also benefit from theknowledge of the refractive index map of the tissue. In fact heterogeneity of refractiveindex inside a sample acts as multiple lenses that locally modify the properties ofthe 3D microscopic imaging, especially in thick samples. Overall, this phenomenoncauses distortions, degrades image resolution and reduces the signal-to-noise ratio;these effects cannot be corrected by global approaches since their effect is local. Ref.[17] proposed to use refractive index map of the sample to model the sample-induceddistortions of the image of a point source and correct their effects by space-variantdeconvolution methods. They assume a linear gradient of refractive index insideeach voxel of measured refractive index values. At this point Snell law allows forray tracking, from which wavefront abberations can be computed. From waveforminformation, a local PSF is calculated and used to locally deconvolve the acquiredimage of the sample. In figure aberrations induced by refractive index heterogeneityare shown and their correction is demonstrated.

1.2 Techniques for measuring refractive indexSeveral techniques have been developed [1] for refractive index measurement of bio-logical samples (fig. 1.3).To measure the effective refractive index of cells, i.e. the refractive index averagedover the whole cell, several techniques have been proposed. Immersion refractometryexploits the intensity contrast between a cell and its surrounding medium using phasecontrast microscopy whereby the cell appears invisible when its effective refractiveindex matches with that of the surrounding medium [18]. A similar method integratedinto a microfluidic chip was demonstrated to measure the effective refractive index ofa single bacterium that is trapped in a microfluidic channel [19]. In addition, severalmicrofluidic chips integrated with various optical techniques were demonstrated tomeasure the effective refractive index, dry mass and water mass of a single livingcell, such as light scattering [20], laser resonant cavity [13], FP resonant cavity [21],grating resonant cavity with optical trap [22], Mach–Zehnder interferometer [23]. Aresolution up to 10−3 was demonstrated. In figure 1.4 is sketched the working principleof optical densitometry and fiber-based resonant cavity. Most of these techniquesmeasure the optical path difference caused by the cell in the medium and decoupleits effective refractive index and size by a differential buffer method [21, 22, 23]. The


Figure 1.2: upper image: sample-induced abberations due to heterogeneous refrac-tive index. A shows an image of a non-distorted image of 0.1-µm bead in-focus and+/- 3 µm defocus while A’ shows an xz section through the center of the 3D image;B panel is the same as A after introducing the bead inside an abberating sample; inC multiple beads are present to show the spatial dependence of abberations; D showsan image of polytene chromatin processed by deconvolution using the symmetric un-aberrated PSF: aberrations are still presents. (Scale bars, 2 µm.) lower image:schematic outline of aberration correction. Measurement of refractive index map byDIC microscopy allows for ray tracking and deconvolution based on ray tracking in-formation. The last figure shows a great improvement in the image compared to thefirst aberrated image. [17]


Figure 1.3: Summary of techniques for measuring refractive index of cells, as foundin [1].

main drawback of this approach is the assumption of a single living cell as a sphericalobject filled with a protein solution. In reality, cells exist in different shapes and otherorganelles exist in the cell especially cell nucleolus that is denser than the cytoplasm.In most cases, minor changes in the concentration and the abundance of variousintracellular organelles are not reflected in the effective refractive index. Therefore,the study of effective refractive index does not provide many details to understandsophisticated cell biological processes.With the development of optical imaging systems a 2D or 3D image of a cell is possi-ble. Localized surface plasmon excitation by using a near-field probe was developed toobtain the 2D refractive index of cell samples on a focal plane by scanning point-by-point [24, 25]. A spatial resolution of several tenths of nanometers with a resolutionof 10−5 was demonstrated [25]. However, it is time consuming since scanning point-by-point is required. One new trend for cell refractive index mapping is quantitativephase imaging. One of the quantitative phase imaging techniques is Digital Holo-graphic Microscopy based on the off-axis configuration whereby two beams, forminga small angle, interferes [26]. One of the two beams passes trough the sample whilethe other is used as reference: the overlap of the two creates an interference figure.This figure is captured by a camera and the phase map of the object beam can beextracted by means of phase unwrapping and background subtraction algorithms.By knowing and making assumption on the shape of the cell, the 2D refractive index


Figure 1.4: (a) average refractive index of cells suspended in a medium measured byoptical densitometry: the transmission of light is maximum when the refractive indexof the medium matches that of the suspended cells; (b) effective refractive index of asingle cell measured by resonant cavity. [1]


profile can be obtained. The demonstrated refractive index resolution was as lowas 10−4 [27]. For the 3D refractive index measurement of a single cell, a series ofphase images are captured at various angles and the 3D refractive index profile canbe reconstructed by algorithms such as filtered back-projection or optical diffractiontomography [28, 29].In next chapter digital holographic microscopy (DHM) will be described in moredetails. In fact a DHM setup was built with the aim of comparing refractive indexmap obtained by Brillouin with a well-known technique.

Chapter 2

Digital Holographic Microscopy forrefractive index measurement

Digital Holographic Microscopy (DHM) is a microscopy techniques that records bothphase and amplitude of the wavefront originating from the sample under inspection.As traditional holography, it exploits interference between a reference beam and abeam transmitted or reflected by an object to create an hologram. The hologram isrecorded by a camera and stored as a matrix of real values, allowing subsequent nu-merical processing. Depending on the optical configuration used for the formation ofthe hologram and on the numerical processing of the acquired holographic image, sev-eral DHM techniques exists. Although sharing the same capability of both amplitudeand phase measurement, different optical configurations and numerical processing arebest suited for different problems that DHM can address like contrast enhancementfor transparent samples, aberration correction, lensless imaging, image refocusing.In this work DHM is used for refractive index measurement: DHM can measurequantitative phase microscopic images and the phase map can then be converted torefractive index map.In this chapter we firstly present the physical principle of holography then, after givinga brief overview on prominent DHM techniques, we will focus on the configurationsused in this work. The problem of 2π phase ambiguity (phase wrapping), that showsup when the optical path length inside the sample is more than a wavelength, will beexplained and phase unwrapping techniques will be presented. Finally the methodfor determining refractive index starting from the phase map will be explained.

2.1 Principle of holography 10

Figure 2.1: Sketch of the physical principle of holography as drawn by Gabor in hispaper (1948) [30].

2.1 Principle of holographyThe idea behind holography was proposed by Gabor1 in 1948 [30]. At the time spher-ical aberration was a limiting factor for resolution of electron microscopy. Since itscorrection was an insuperable task, he proposed to radically change the way imageswere formed and analysed. Instead of trying reducing the spot size of the focusedelectron beam to increase resolution, he suggested to use a beam that is even largerthan the object. In this way, after the object, two waves are formed: a primarywavefront that is not affected by the object and a secondary wavefront that containsinformation on the object (fig. 2.1). If a photographic plate is placed at some dis-tance, the interference between the primary and secondary beam can be registeredand he demonstrated that the interference pattern contains information on both am-plitude and phase of the secondary wave: after developing the photographic plate itis illuminated with an exact replica of beam used to create the hologram; the lighttransmitted by the photographic plate contains information about amplitude andphase of the original beam diffracted by the object and, in certain conditions, it isjust a replica of that beam. He called this technique holography, from the Greekwords holos (“whole”) and graphe (“writing” or “drawing”), to underline that thewhole wavefront is reconstructed in both phase and amplitude.To understand the principle behind the reconstruction of the whole wavefront inDHM, let’s calculate the intensity that is registered by the camera. Let x and y bethe spatial coordinates in the detector plane. Ur(x, y) is the wavefront of the referencebeam while Uo(x, y) is the wavefront of the beam that interacted with the object; bothare complex quantities that encode the amplitude and the phase of the fields in thedetector plane. The detector can measure only the intensity of the resulting field that

1In 1971 he was awarded the Nobel Prize in Physics “for his invention and development of theholographic method”.

2.2 Overview on holographic configurations 11

is the sum of Ur and Uo:

I(x, y) ∝ |Uo + Ur|2 = |Uo|2 + |Ur|2 + UoU∗r + U∗oUr (2.1)

An important note to the previous expression is that it is valid only if the two beamsare coherent, otherwise the total intensity would be the sum of |Uo|2 and |Ur|2 withoutthe interference terms. Hence a laser radiation must be used to create an hologram.In fact the useful information is contained in the interference term UoU

∗r ; since Ur is

a beam used as reference, its phase and amplitude are known and, measuring UoU∗r ,Uo can be inferred. The term U∗oUr is known as twin image while |Uo|2 + |Ur|2 is theDC component.Basically two problems must be addressed, in order to measure the wavefront pro-duced by the object:

• the term UoU∗r must be isolated from the other terms in equation 2.1

• Uo is the object wavefront in the detector plane that, in general doesn’t coincidewith the object plane; thus, once Uo is determined, it must be related to thefield in the object plane

Several strategies can be adopted to overcome the previous difficulties. As far as twinimage removal, common strategies are spatial or spectral filtering and acquisitionof multiple phase shifted photograms. For wavefront reconstruction, in analogicalholography a duplicate of the reference beam is used in the reading process; instead, inthe digital holography, numerical diffraction is computed. Fresnel diffraction is widelyused in numerical diffraction, if its hypothesis are met, since it is computationally veryefficient.Some strategies will be briefly explained in the following section. Afterward moredetails will be given on the configuration adopted in this work, explaining why it waschosen among the others.

2.2 Overview on holographic configurationsSince holography is an interferometric technique, the core of its hardware implemen-tation is an interferometer. The Mach-Zehnder configuration is generally used forimaging in transmission while the Michelson is used for reflective samples. Also con-figuration having a single arm were reported: Gabor configuration is an historicallyrelevant example. What also differentiates the various configurations is the shape(plane or spherical wave) of the wavefront that is used for illuminating the sampleand the shape of the one used as a reference beam, the angle between the two beamsand the distance between the object and the hologram plane.Below some basic configurations are presented [31, 32].


Figure 2.2: Gabor holography. (a) Recording by superposition of the reference waveand its scattered component from a point object, and (b) reconstruction of a pointimage (-1 order) and its defocused twin (+1 order). [32]

Gabor holography This is the simplest configuration, it doesn’t require any opticalelement and it follows the original idea by Gabor. A collimated beam illuminatesthe object. The light diffracted by the object interferes with the light that isnot disturbed by it. In order to have a reference beam that is “clean”, theilluminating beam must be much larger that the object (fig. 2.2). When readingthe hologram, the DC terms are not disturbing since they form a constantbackground intensity (the scattered field is much weaker than the referencefield). The twin image instead overlaps to the desired image; anyway, recallingeq. 2.1, the twin image is the complex conjugate of the real image, thus it formsat a position that is specular to the real image with respect to the photographicplate (fig. 2.2). Therefore if the object is small even at relative small distancesthe twin image is defocused and its contribution to the overall image can benegligible.Due to its characteristics, Gabor holography is particularly useful for particleor thin fiber image analysis.

In-line holography The inline configuration is very similar to the Gabor configu-ration, the only difference being a second beam that is used to illuminate thesample (fig. 2.3). The main advantage is that in this configuration there areno restriction on the size of the object with respect to the field of view. On theother hand it is necessary to remove the DC terms and the twin images. Severaltechniques can be used; one of that is phase shifting that will be explained laterin this section.

Fourier holography This configuration exploits the Fourier transform property ofa lens: the field at the back focal plane of a lens is the Fourier transform ofthe field at the front focal plane. A shown in figure 2.4a, the object is placed


Figure 2.3: In-line holography. (a) In-line superposition of object and referencebeams, and (b) reconstruction of superposed zero-order and twin images. [32]

at the front focal plane of a lens and a reference beam is focused in the sameplane just beside the object; the hologram is registered in the back focal plane.In the reading process the holographic plate is at the focal plane of the lensthat performs the inverse Fourier transform, reconstructing the object. Bothof the twin images are in focus at the focal plane of the imaging lens and thezero-order is a small intense spot (fig. 2.4b).In the digital holography the reading process is reduced to a numerical Fouriertransform.The Fourier holography can also be implemented in a lensless configuration(fig. 2.4c): the object and the focused reference beam are in the same plane, asbefore, but there is no lens between this plane ad the camera. It can be shown,using Fresnel diffraction theory, that the term UoU

∗r in the interference figure is

basically proportional to the Fourier transform of the object beam.

Fresnel holography In this configuration the object is at a finite distance from theholographic plane and the reference beam is a plane wave that is at an angle withrespect to the object (fig. 2.5a). If the hologram is reconstructed analogically,illuminating with a plane wave the photographic plate, the image is formed atthe object position, while the twin image is at the mirror position with respectto the photographic plate (fig. 2.5b). If the hologram has to be reconstructeddigitally, a numerical Fresnel diffraction is carried out.To reconstruct the DC terms and the twin images without aliasing, the distancebetween the screen and the object must be larger than 3X2

0Nxλ

where X0 is thescreen size and Nx the number of sampling points (the same relation must holdin the y direction).

Both the Fourier and Fresnel configuration can be implemented without lensestherefore are ideal to be used in spectral regions where lenses doesn’t exist likex-ray.


Figure 2.4: Fourier and lensless Fourier holography. (a) Fourier hologram recordingusing a lens; (b) reconstruction by Fourier transform, represented with the Fourierlens; and (c) lensless Fourier hologram recording. [32]

Figure 2.5: Fresnel holography. (a) Recording by off-axis superposition of the objectand reference waves, and (b) reconstruction of separated zero-order and twin images.[32]


Figure 2.6: Phase-shifting digital holography. PZT: piezomounted mirror for modu-lation of reference phase. [31]

Phase-Shifting Digital Holography The strategy of phase shifting can be appliedto most digital holographic configuration and it allows to solve the problem oftwin image and removal of DC terms.The complex reference and object fields can be written as Uo(x, y) = Eo(x, y)eiϕ(x,y)

and Ur = Ere−iψ. The hologram intensity will be in the form:

Iψ(x, y) = E2o(x, y) + E2

r + 2Eo(x, y)Er cos [ϕ(x, y) + ψ]

As shown in figure 2.6, the phase ψ of the reference beam can be controlledwith a piezo mirror. Thus multiple holograms can be acquired varying ψ. Iffour different values are used, ψ = 0, π2 , π,

32π:

I0 = E2o + E2

r + 2EoEr cos (ϕ)

Iπ/2 = E2o + E2

r − 2EoEr sin (ϕ)Iπ = E2

o + E2r − 2EoEr cos (ϕ)

I3π/2 = E2o + E2

r + 2EoEr sin (ϕ)Combining those four holograms the amplitude and phase of the object beamcan be retrieved:

Eo(x, y) = 14Er|(I0 − Iπ) + i(I3π/2 − Iπ/2)|

ϕ(x, y) = arctan(I3π/2 − Iπ/2I0 − Iπ

)Based on this principle, it is also possible to reduce the number required pho-tograms to two or even one at the expense of the spatial resolution.

2.3 Image plane and off-axis DHM configuration 16

The previous discussion assumes that the only quantity that varies betweensubsequent expositions is ψ. Of course in a real setting there is noise and alsoother quantities are varying; this led to incomplete cancellation of twin imageand DC terms. Several statistical and optical approaches have been proposedto reduce this problem.

2.3 Image plane and off-axis DHM configurationIn this work it was used an off-axis, image plane DHM configuration for the refractiveindex measurement.In the image plane configuration the object plane is close to the hologram plane.In the simple case in which the object is physically close to the hologram plane,the magnification is 1 thus is not ideal for microscopy. An improvement consists inintroducing a lens after the object and register the hologram near the conjugate plane.In this way the image will be magnified, however the phase will contain the curvaturedue to the imaging lens. Instead if a 4-f system is used to magnify the object fieldalso the phase is magnified without distortions.In image plane digital holography numerical diffraction is not required to reconstructthe object, since its image is already at the holographic plane. Other advantagesof this configuration are the non disturbance of light scattered by portion of thesample that are out of focus, the independence of resolution on the sensor size. Thesecharacteristics allow high resolution images, the limit being the NA of the objectiveand the sampling frequency of the camera [33].Off-axis is a configuration that allows to separate the image from the DC terms andthe twin image and is widely used due to its simplicity and efficiency in a digitalimplementation. The starting point is equation 2.1 describing the interference be-tween two generic waves. We now assume that the object wave is a function of xand y (coordinates on the camera plane, as shown in fig. 2.7), Uo(x, y) while thereference beam is a plane wave Ur = Are

i(kxx+kyy+kzz). If an angle is formed (inset infig. 2.7) between the reference beam and the z axis, that correspond to the directionof propagation of the sample beam, then kx 6= 0 and ky 6= 0. Equation 2.1 becomes:

I(x, y) ∝ |Uo|2 + A2r + UoAre

−i(kxx+kyy+kzz) + U∗oArei(kxx+kyy+kzz)

Fourier transforming the previous expression with respect to x and y one obtains:

I(r, s) ∝ F (|Uo|2)+A2rδ(0, 0)+ArF (Uo)⊗δ(r−kx, s−ky)+ArF (U∗o )⊗δ(r+kx, s+ky)

The effect of the angle (kx 6= 0 and ky 6= 0) is to move in the Fourier plane the objectfield F (Uo) and its twin image F (U∗o ). If the angle is big enough the two images arecompletely separated one from the other and from the intensity image F (|Uo|2) that


Figure 2.7: Schematics of off-axis configuration for DHM. The inset shows the anglebetween reference and object beam. [34]

it is centered at the origin of the Fourier plane. The images are completely separatedwhen the bandwidth of each image doesn’t intersect with the adjacent one. The upperlimit for the angle is imposed by Nyquist theorem: the sampling frequency (given bythe pixel of the CCD) must be twice the maximum spatial frequency present intothe hologram to avoid aliasing. Considering that the DC term |Uo|2 is the absolutesquare of the image terms Uo and U∗o , its size in the Fourier space will be twice asbig as the size of image terms. Therefore optimal separation is achieved when thebandwidth of Uo is one fourth of the Fourier plane (fig. 2.8). It is clear that thefiltering process reduces the bandwidth to a fourth of the original hologram thus itworsens the resolution of the same factor.The reconstruction of the complex object field proceeds as follow: after acquiring thehologram, its numerical 2-dimensional Fourier transform is computed. The Fouriertransform is composed of three spots corresponding to DC terms, the image andits twin counterpart (insets in figure 2.9). The spot corresponding to the image isselected with a filter that could be as simple as selecting an appropriate area aroundit and zeroing out everything else. After filtering, a reverse Fourier transform iscomputed to reconstruct the complex image. The PSF of the reconstructed imageis determined by the shape of the filter; for a circular aperture the PSF is a Besselfunction that presents fringes on the sides of the central peak. To reduce fringing amask can be used in filtering, for example a gaussian mask will produce gaussian PSFin the reconstructed image. As already mentioned, the size of the filter determines the


Figure 2.8: Optimal spectral configuration of the Fourier transform of an hologramfor best separation of the image from its twin counterpart and DC terms. [35]

Figure 2.9: Filtering in the Fourier plane with varying filter sizes. [32]

2.4 Phase unwrapping 19

resolution: the larger the filter, the finer the resolution. Anyway if the filter includespart of the DC term some artifacts are shown in the final image (image on the rightof fig. 2.9).

2.4 Phase unwrappingAll the methods previously described allow to obtain the complex field Uo that con-tains information on both phase and amplitude

Uo(x, y) = Ao(x, y)eiϕ(x,y) (2.2)

from which one can calculate amplitude and phase:

Ao = |Uo|

ϕ = arg(Uo)Anyway, while inverting relation 2.2 for amplitude is a trivial task, things are notthat simple for the phase. In fact phase is affected by a problem known as wrapping:the argument of a complex number is a number between 0 and 2π, that correspondto an optical path of one wavelength inside the sample; if the optical path is longer,the phase will be greater than 2π and the measured value will be the actual phasemodulo 2π. Therefore the correct value for the phase must be reconstructed aftertaking the argument.

2.4.1 One-dimensional phase unwrappingLet’s first consider the case of one-dimensional phase. In figure 2.10 the argumentof function e20πi·(x3−x2) is shown, as an example. In the upper image it is shown thephase wrapped between 0 and 2π; in the lower image it is shown the phase afterunwrapping. In a single dimension the unwrapping process simply consists of addingor subtracting 2π every time a jump of more than π is found.If the signal is noisy, there may be jumps in phase that are not due to the wrappingbut to the noise; if those jumps are higher than π, the unwrapping algorithm will add2π affecting also the sample on the right of the fake phase wrap (fig. 2.11).Also under-sampling of the signal can leads to problem in the unwrapping process.In fact, let’s consider a sinusoidal signal (fig. 2.12 (a) original and (b) wrapped)

x(t) = 10 sin(10t) 0 ≤ t ≤ 1

In order for the unwrapping algorithm to correctly reconstruct the phase, the phasedifference between two subsequent points must never exceed π. If we approximatethe difference with the derivative:

∆ϕ ≈ dx(t)dt

∆t = 100∆t cos(10t)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

1

2

3

4

5

6

phas

e (r

ad)

wrapped phase

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

-10

-8

-6

-4

-2

0

phas

e (r

ad)

unwrapped phase

Figure 2.10: Example of 1D phase wrapping, arg(e20πi·(x3−x2)).

Figure 2.11: (a) Original noisy signal, (b) the phase wrapped signal, (c) the phaseunwrapped signal. [36]


Figure 2.12: Unwrapping a slightly under sampled discrete signal where the numberof sampling points is 31, just below the minimum value for correct reconstruction ofthe signal. [36]

Since ∆t = 1/N , where N is the number of sampling points, assuming the maximumvalue of 1 for the cosine, the following relation must hold:

100N

< π

that is to sayN >

100π

= 31.83

In figure 2.12c it is shown the case where the number of sample points is just belowthis limit.

2.4.2 Two-dimensional phase unwrappingUnwrapping in two dimensions poses further problems. The first solution to 2Dunwrapping that may come to mind is to unwrap all the rows first and then unwrapall the columns (or viceversa). This indeed solves the problem and the unwrappedphase doesn’t change if the unwrapping order of rows and columns is changed. Butthat is true only for an ideal phase image: in case noise or under-sampling is presentthis algorithm is very unreliable. Since, as discussed in the one-dimensional case,errors on fake phase jumps compromise all the data after the jump, unwrapping errorsin one direction propagate in the other direction. It is clear that errors are amplifiedby this process and the result will depend on the unwrapping order of columns androws [37].This makes phase unwrapping a complex problem and several algorithms have beendeveloped that address specific problems or phase topologies [38]. The approaches totwo dimensional phase unwrapping can be divided into three categories of algorithms[39].


path-following This class deals with the problem of two dimensional unwrappingby choosing a path and performing a one dimensional unwrapping along thatpath; based on how the path is selected, three main categories exist:

path-dependent these algorithms use an a priori-defined search path; theyare usually fast but are unable to handle noisy images because of the fixeddata evaluation order;

quality-driven these algorithms are similar to path-dependent algorithms butthe unwrapping order is established by the pixel reliability having betterperformances on noisy images;

residue-compensation these algorithms generate cuts in the integration do-main so as to connect two or more opposite sign residues; although thisprocedure ensures that there is no rotational component in the phase field,the unwrapping quality depends on the cut selection strategy; the Gold-stein’s algorithm, that is the one used in this work and it is described inmore details in appendix A.2, belongs to this category.

region The phase image is divided into homogeneous regions that are then assem-bled in order to reduce the phase difference at the interfaces; several strategiescan be adopted to select the areas: the region family generates homogeneousareas based on phase gradients, the tile family divides the original image intoa grid of smaller areas that are unwrapped either by a simpler algorithm (usu-ally a path-dependent one) or recursively by calling the same algorithm, theminimum discontinuities algorithm creates paths in the phase field by follow-ing discontinuity curves and extending them to form complete regions. All ofthem are known to be robust even if the image division process can be difficult,computationally intensive, and error prone.

global The problem of unwrapping is formulated in terms of minimization of a globalfunction; those algorithms are usually computationally intensive.

2.4.3 Optical phase unwrappingOptical phase unwrapping allows to extend the maximum optical path after whichphase wrapping happens. This comes at the price of acquiring the same hologramwith two different wavelengths. [32]Let’s call λ1 and λ2 (λ2 > λ1 for definiteness) the two wavelengths used for hologramacquisition. The corresponding phase profiles are ϕ1(x, y) and ϕ2(x, y) and vary from0 to 2π corresponding to a single wavelength long optical path. Now the differencebetween the two phase profiles is calculated, ∆ϕ = ϕ1−ϕ2, and 2π is added wherever∆ϕ < 0 leading to a new phase map Φ′12 that ranges from 0 to 2π and has an effective

2.5 Refractive index measurement 23

(or synthetic) wavelength of:Λ12 = λ2λ1

λ2 − λ1

The new maximum range for optical thickness before phase wraps is given by Λ12and, for small enough difference between the two wavelengths λ2−λ1, it can be madelarge enough to cover the whole thickness of the object. However if the original phasemaps contain noise, it will be amplified by the same factor as the synthetic wavelengthmagnification. To reduce the noise to its original level, one can use the new phasemap Φ′12 as map to unwrap the single wavelength phase map. That is:

Φ12 = ϕ1 + 2π int[

Φ′12Λ12

λ1

]

This method works if the amplified noise doesn’t exceed the original wavelength. If thenoise is more excessive or a larger synthetic wavelength is needed, more wavelengthscan be used.

2.5 Refractive index measurementIt is well known that the phase delay acquired by beam going through a mediumof refractive index n and length l is ∆ϕ = 2π

λnl, thus if ∆ϕ is measured and l is

known, the refractive index can be calculated. This is the basic idea that can easilybe extended to the case of variable refractive index n(x, y, z), following Ref. [40].Let’s assume that the beam is propagating in the z direction and it goes through asample, that has a thickness t(x, y). At the exit of the sample it will have travelledan optical path ` that is given by:

`(x, y) =∫ t(x,y)

0

2πλn(x, y, z)dz

Note that the previous equation is valid only if light is not deflected by the sample.Otherwise the phase at the exit of the sample will have contributions also from pointsthat are not directly above a specific point, i.e. the phase at a point (x, y) at the exitof the sample will depend on points (x′, y′, z) with x′ 6= x and y′ 6= y. The amountby which the light is deflected depends on the variations of refractive index inside thesample and between the sample and the medium, thus all the subsequent discussionis valid only for samples that have a refractive index close to that of the medium.Biological samples respect these assumptions because they are mainly made up ofwater and exhibits a refractive index close to 1.33 with variations of about 0.1.If the sample is immersed in a medium of refractive index nm and thickness D, theprevious expression becomes:

`(x, y) =∫ t(x,y)

0n(x, y, z)dz + nm[D − h(x, y)] = (n(x, y)− nm)h(x, y) + nmD


Figure 2.13: x-y slices of the reconstructed refractive index map of a CA9-22 cell (fig.6 in [41]).

wheren(x, y) = 1

t(x, y)

∫ t(x,y)

0n(x, y, z)dz

is the mean value of the sample refractive index along its thickness and it is calledintegral refractive index.From previous equations it is clear that, in a single shot, one can measure onlythe integral refractive index, assuming the thickness of the sample is known. Thethickness needs to be determined with other techniques or some assumptions must bemade. For example if the sample can be well approximated by a sphere, the thicknesscan be determined from the radius that can be measured from the 2D amplitudeimage. Some kind of cells can be well approximated with a sphere while for othersfurther analysis is needed to obtain a thickness profile.This approach can be extended for 3D imaging. In 2006 Ref. [42] demonstratedthat 3D reconstruction of refractive index is possible using DHM. They acquired 90images of a 30 µm diameter yew pollen grain from different orientations, coveringan overall angle of 180°. The representation of the data as a function of the angle isknown as a sinogram. The 3D signal ∆n(x, y, z) was reconstructed from the sinogramsby a filtered backprojection algorithm. Later studies examined more reconstructionalgorithms.In figure 2.13 it is shown the 3D refractive index map of a cell [41].

Chapter 3

Experimental setup for DigitalHolographic Microscopy

In the previous chapter we discussed how Digital Holographic Microscopy (DHM) canbe used to obtain a refractive index map of a biological sample. In this chapter we willdescribe the DHM setup built at this end, giving its characterization and presentingexperimental results that prove the correctness of refractive index measurement.

3.1 DHM setupA schematic of the setup is shown in figure 3.1. The core of the setup is a Mach-Zehnder interferometer: after spatial filtering, laser light is sent to a beam splitterthat produces two beams, one for illuminating the sample and the other to be usedas reference beam. The sample is fixed on a microscope assembly and is imaginedonto the camera by means of a 4f system1 whose first lens is a microscope objective.The reference beam is magnified and combined with the sample beam by means ofa second beam splitter. The two beams form an angle that allows to separate theimage from DC terms and twin image2.More in details, the laser is a solid state green laser at 532 nm and it is the sameas the one used for Brillouin measurements. The beam emitted by the laser has adiameter of about 1 mm. A spatial filter is used to produce a clean intensity profile.The first aspheric lens of the spatial filter has a focal length of 11 mm. The expectedAiry disc for the focused spot is about 14 µm, thus a 15 µm pinhole is used. Afterfiltering, the light is collimated by an aspheric lens of 15 mm of focal length. Theexpected beam diameter is 1.36 mm. A 50:50 beam splitter is used to separate thebeam that illuminates the sample from the beam used as reference.

1see image plane configuration in sec. 2.32see off-axis configuration in sec. 2.3

3.1 DHM setup 26

Spatial filter

Beam

expander

Microscope

assembly

f

Figure 3.1: Schematic of the DHM setup.

3.1 DHM setup 27

Figure 3.2: Picture of the DHM setup.

The beam used as reference is magnified by a factor 5.9 by a beam expander madeup by two lenses of focal length 25.4 mm and 150 mm respectively. In this way thereference beam will be approximately 8 mm in diameter, so that the central region ofthe sensor (9mm x 6.7mm) is illuminated with uniform intensity.The sample is fixed on a microscopy assembly and it is illuminated from the top.The beam line is raised by means of a mirror at 45°, then other two mirror mountedat 45° on a cage system, held up by a pillar, direct the light onto the sample (fig.3.3). After going down and passing through the sample, the light is redirected, byanother mirror at 45°, to an optical line that is at the same hight as the referencebeam. Magnification of the object field is achieved by means of a 4-f system madeup by an objective lens (0.75 NA) and a 400 mm focal length lens. The camera isplaced in the back focal plane of the latter lens, after a beam splitter that recombinesthe reference and object beam. The 4-f system produces a magnified replica (bothin phase and amplitude) of the beam after the sample. The focus of the image isadjusted by means of the knob on the microscope assembly.In figure 3.2 it is shown a picture of the setup, while in figure 3.3 is shown a detailof the cage system, supported by the pillar, used to illuminate the sample from thetop.

3.1 DHM setup 28

Figure 3.3: Detail of the cage system used to illuminate the sample from the top.

3.1.1 ResolutionThe optical resolution of DHM is determined by the numerical aperture of the objec-tive lens that collects the light after the sample. In our case a NA of 0.75 correspondsto a resolution of about 0.61λ/NA = 0.4 µm. In figure 3.4 is shown the image of aUSAF resolution test target. The thinnest lines (group 7 element 6) have a widthof 2.19 µm, thus 5 times bigger than the expected resolution and indeed they areresolved.The image of the resolution test target allows to calculate the magnification of theoptical system, knowing the pixel size that is a square of 6.45 µm. After determiningthe position (in pixels) of the edge of each line, a linear regression allows to calculatethe width of a line, thus to calculate the magnification. This measurement was donefor element 1, 2 and 3 of group 7, obtaining a mean magnification of 43.1. This valueshould be compared to the expected magnification: the nominal magnification of theobjective lens is 20x, using a 180mm tube lens3, thus with a 400mm tube lens itbecomes 44.4x.A rule of thumb states that for optimal sampling the optical point spread functionshould be about 2 pixels in size. Considering the pixel size of 6.45 µm and themagnification of 43.1, each pixel correspond to a sampling size of 6.45/43.1µm =

3Each manufacturer uses a different convention. The convention used by Olympus is 180mm tubelens (https://www.microscopyu.com/microscopy-basics/infinity-optical-systems).

https://www.microscopyu.com/microscopy-basics/infinity-optical-systems

3.1 DHM setup 29

Figure 3.4: Image of a USAF resolution test target, taken with the DHM setup. Thethinnest lines (group 7 element 6) have a width of 2.19 µm.

0.15µm, so the PSF (0.4µm) is sampled by 2.6 points.

3.1.2 Phase accuracyThe phase accuracy was calculated as described in [43]. Ten frames were acquiredwithout any sample. The phase was calculated using the procedure of Fourier filteringand phase unwrapping, as explained in the previous chapter. For noise measurementonly the central part of the phase image was taken into account (in our case a borderof 1/5 of the whole image was removed); this is done to prevent the influence of bordereffects due to the discontinuity introduced by the windowing of the hologram whenit is processed by the FFT calculation. The mean of all frames was subtracted fromeach frame to remove any phase pattern that is caused by optical imperfections ormisalignment and thus it is not fluctuating in time. After this procedure, ten framesare obtained in which only the phase pattern that is not constant in time is present.The standard spatial deviation of each frame is calculated, obtaining ten values ofphase noise. These values are finally averaged to obtain a single value for phase noise.A Labview program was written to automatize the procedure.In figure 3.5 is shown the measured phase noise versus the exposure time. The opticalpower on the sample arm was about 8 mW while, to match the two intensities, thepower on the reference arm was about 0.3 mW. On the same graph the averagenumber of photons per pixel is also plotted; it is calculated considering a full well

3.2 Experimental data 30

0

2000

4000

6000

8000

10000

12000

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0 5 10 15 20 25 30 35

mea

n p

ho

ton

s/p

ixel

ph

ase

no

ise

(rad

)

exposure time (µs)

Phase noise

phase noise (rad) mean photons/pixel

Figure 3.5: Phase noise versus exposure time (blue dashed curve). The orange linemaps the exposure time to the mean number of photons per pixel.

capacity of 18000 e− and a quantum efficiency of 0.55, as declared by the constructor4.Oscillations in the phase noise have not physical significance but are due to randomfluctuations since they change shape in repeated measurements. The phase noise isabout 5 times bigger than the shot noise level, as simulated in Ref. [43]. Anyway, asfound in Ref. [44], the noise level in DHM depends also on the size of the hologramand on the fringe contrast and these factors could account for the discrepancy betweenshot noise and measured noise level.The minimum phase noise was 0.04 rad corresponding to an accuracy of ∼ λ/150 ≈3nm.

3.2 Experimental dataMicrospheres of known material were used to verify the correctness of DHM phasemeasurement. If a sphere of refractive index ns and radius R is immersed in a mediumof refractive index nm, it will induce a phase delay of

2πλ

(ns − nm)2√R2 − r2 (3.1)

where r is a coordinate that measures the distance from the center of the sphere. Byfitting the previous expression to the phase image of a sphere acquired with DHM,

4https://www.qimaging.com/products/datasheets/exi_blue.pdf

https://www.qimaging.com/products/datasheets/exi_blue.pdf


Figure 3.6: Screenshot of the Labview program written for real time DHM imaging.

both R (that can be compared to the value declared by the manufacturer) and ns canbe measured.The spheres had a refractive index close to 1.5. As a medium it was used an oil ofrefractive index 1.5181 @546nm. It is necessary to have a small difference betweenthe refractive index of the sphere and that of the medium because otherwise the phasedelay would be too high and some phase jumps could be lost in the phase unwrapping,leading to a wrong value for refractive index.

3.2.1 Data acquisition and analysisA Labview program was written to acquire real time DHM images (fig. 3.6). Realtime images allowed to center the sample in the field of view, by using an x-y mo-torized stage mounted on the microscope assembly, and to adjust the focus. Focusis particularly critical since, due to its spherical shape, the sphere acts as a lens andphase rapidly evolves on a spatial scale of few microns. Five frames were acquiredfor each sphere, in order to make an average. The subsequent analysis was done inMatlab. Figure 3.7 shows the steps in the analysis. From the acquired hologram (a)the 2-dimensional FFT is computed (b). Then it was selected the biggest area aroundthe first order that doesn’t intersect with the zero order. By means of an inverse FFTthe phase of the acquired hologram was calculated (c); jumps of 2π due to phase


a) b)

c) d)

e) f)

Figure 3.7: Steps in the analysis of the DHM images of spheres for measuring therefractive index. (a) hologram (b) fourier transform (c) phase image (d) unwrappedphase image (e) phase image after background removal (f) fitting of the function2πλ

(ns − nm)2√R2 − r2 to a single sphere.

wrapping are clearly evident. The unwrapping algorithm (sec. 2.4) allows to elimi-nate those discontinuities (d). The background shows a gradient, that may be causedby some non uniformities either of the medium or the Petri dish that contains thespheres or by aberrations of the wavefronts. To remove it, a second order polynomialfit is performed on the whole image; due to the low order of the polynomial, the fittedsurface can’t follow the phase of single spheres but only the phase variations due tobackground. Therefore fitting is an effective way for background removal (e). At thispoint a “clean” phase image is obtained. Finally the single spheres are selected andfitted with equation 3.1 (f), obtaining the value for both the radius and the refractiveindex.


Material # spheres Tabulated n Measured n std nPolystyrene 16 1.5983 1.5955 0.0023PMMA 49 1.4934 1.4926 0.0004Glass 36 1.5261 1.5207 0.0008

Table 3.1: Refractive index measurement of microspheres of 3 different material usingDHM. “# spheres” is the number of spheres used for measurement. Note that “Tab-ulated n” refers to the tabulated value of the material and there may be variationswith respect to the actual index of the microsphere.

3.2.2 ResultsIn table 3.1 are summarized the measured refractive indices for 3 different materials:the PMMA and glass spheres have a diameter of 30 µm while the polystyrene oneshave a diameter of 20 µm. Sine we can assume that the refractive index is exactly thesame for each sphere of the same material, the standard deviation is an estimationof the accuracy of the technique. The values in table 3.1 suggest that a very highaccuracy can be achieved for refractive index measurement using this technique.An important note to table 3.1 is that the refractive index for polystyrene sphereswas obtained by adding a 2π shift ad hoc. Polystyrene is the material that has thehighest difference in refractive index from the medium among the three. Indeed if theoptical path of the sample varies more than a wavelength on a spatial scale that islower than the spatial sampling frequency, the unwrapping algorithm will miss somejumps and fail in the correct reconstruction of the phase. The problem may arise nearthe border of the sphere, where there is a steep variation in thickness. In figure 3.8it is shown the phase image of a polystyrene sphere before and after the unwrapping:it can be seen that at the transition between the sphere and the medium the phaseincreases instead of decreasing; this can indicate a phase jump that is missed. Thereis another assumption behind equation 3.1 that could be violated by polystyrenespheres: it is possible to write ∆ϕ = 2π

λ(ns − nm)t (where t is the thickness of the

sphere at a particular point) only if light is not deflected by the sphere but only itsphase changes. Obviously this is not true in general because a sphere is actually alens. But for small difference in refractive index between the sphere and the medium,the lensing effect is negligible.


Figure 3.8: Phase image of a polystyrene sphere before and after the unwrapping.Note that the transition between the sphere and the medium is not smooth and itmay be possible that in phase unwrapping a 2π-jump is missed.

Chapter 4

Confocal Brillouin microscopy forrefractive index measurement

Confocal Brillouin Microscopy is a confocal microscopy technique that exploits Bril-louin scattering for obtaining information about acoustic, thermodynamic and vis-coelastic properties of a material [45]. Brillouin scattering is the interaction betweenphotons and thermal phonons intrinsically present in materials at room temperature.The scattered photons carries information about the refractive index at the pointwhere interaction occurred. Thus a slightly modified Brilloun microscope can beused for local measurement of refractive index. The local measurement is the nov-elty of this work since, although Brillouin scattering was exploited in the past forrefractive index measurement [46], it was never used for 3D mapping of refractiveindex.In this chapter the bases of Confocal Brillouin microscopy are examined: firstly thephysical principle is briefly elucidated, then challenges for using this technique in bio-logical samples and adopted solutions are presented. The chapter will finally presenthow it is possible to measure refractive index.

4.1 Brillouin scatteringBrillouin scattering is a scattering phenomenon, named after Léon Brillouin, betweenmaterial waves inside a medium and light. In any medium, local density and pressureintrinsically fluctuate due to fundamental thermal energy. Such density and pressurefluctuations generate periodic and traveling refractive index variation that can bethought as microscopic sound waves, hence acoustic vibrations or phonons. At amicroscopic level then the medium is equivalent to a traveling diffraction grating thatpropagates in the medium at the velocity of sound. As a result, when light entersthe material it can be diffracted in a different direction in a similar fashion to Bragg-diffraction occurring in crystals and with different frequency due to an exchange of

4.2 Technological challenges and solutions 36

energy similar to the frequency shift occurring due to Doppler interactions [47].The model used to calculate the frequency shift between the incident light (denotedwith the subscript i) and the scattered light (denoted with the subscript s) is a photon-phonon interaction. From conservation of energy and momentum (characteristicsrelated to the phonon are denoted with the subscript B)

ωB = ωi − ωs

~kB = ~ki − ~ksit follows that the Brillouin frequency shift Ω is

Ω = 2vλn sin θ2 (4.1)

where v is the speed of sound in the medium, λ is the wavelength of incident light invacuum, θ is the angle between the directions of incident and scattered light and n isthe refractive index of the medium at the interaction point.The measurement of Brillouin shift can be used to infer mechanical properties ofmedia, encoded in the sound velocity1. But, due to very low cross section in thespontaneous regime and low frequency shift (∼GHz) of this scattering phenomenon,an high extinction ratio and sub-GHz resolution is needed: in the past different ap-proaches were used that imply the measurement of each spectral component sequen-tially, making the acquisition of a single spectrum very slow (up to hours, dependingon the insertion loss of the instrument). Scarcelli and Yun [45] proposed the use ofa multistage virtually-imaged phased array (VIPA), discussed in section 4.2.1, thatallows the reduction of acquisition time to less than a second, while still maintaininghigh extinction ratio with sub-GHz resolution.

4.2 Technological challenges and solutionsAs any confocal technique, in standard Brillouin microscopy the pump laser is focusedonto the sample. After interacting with thermal phonons, light is scattered in anydirection and must be collected2 in order to measure the Brillouin frequency shift; acommon approach is epidetection, in which the backscattered light is collected by thesame objective that is used to focus the pump laser. To filter out specular reflection,the polarization of light can be exploited: reflection preserves polarization of lightwhile scattered light has random polarization; therefore, using a quarter waveplateand a polarizing beam splitter, unwanted reflection is ideally totally rejected while

1v =√

M ′

ρ where M ′ is the real part of the longitudinal elastic modulus of the material and ρ isthe density of the material.

2The NA of collecting lens determines the collecting volume, thus the spatial resolution.


t

R=100%

R=95%

AR (R=0)

Virtual images

incident beam

beam waist

VIPA

Figure 4.1: Schematization of working principle of a VIPA (https://commons.wikimedia.org/wiki/File:VIPA_Prinzip.svg).

signal is reduced by 50%. However, even after removal of reflected light, there isstill a background, mainly due to elastic scattering, that is orders of magnitude moreintense than the signal, therefore the spectrometer must have an high extinction ratio.To achieve these characteristics a multistage VIPA spectrometer can be used. Thespectrometer converts the frequency spectrum of the signal into a spatial patternhaving a size of few hundreds microns and very low intensity. The detector must becapable of detecting such low intensity (hundred of photons) while still maintainingspatial resolution: EM-CCDs have such performances.In this section we explain how a multistage VIPA spectrometer reaches such an highspectral resolution and high extinction ratio and the basic working principle of anEM-CCD.

4.2.1 Multi-stage VIPA spectrometerA virtually-imaged phased array (VIPA) is a special type of Fabry-Perot etalon withthree distinct coatings: the front surface is highly reflective (nearly 100%) with theexception of a narrow strip with anti-reflection coating used as entrance for the light,while the back surface has a partially transmissive coating. As shown in figure 4.1,light is focused on the VIPA entrance with a small angle, so that multiple reflectionsbetween the two surfaces can exit the VIPA at different positions. Each subsequentbeam has a precise increase in phase and fixed lateral displacement and behaves as avirtual image of the sources, hence the name virtually-imaged phased array. Beamswill interfere constructively at an angle that depends on the wavelength.There are several parameters that define the performance of a VIPA [48].

https://commons.wikimedia.org/wiki/File:VIPA_Prinzip.svg

https://commons.wikimedia.org/wiki/File:VIPA_Prinzip.svg


optical thickness OPD: for a solid etalon this is OPD = 2n · t · cos Θ, where nis the refractive index of the VIPA, t is its thickness, and Θ is the angle fromnormal within the VIPA, as in figure 4.1; the OPD is related to the free spectralrange FSR by the equation FSR ≈ c/OPD.

reflectance of back surface: higher reflectance implies higher resolving power butlower throughput.

incident angle Θ: smaller angles increase the angular dispersion, but the lower limitdepends on the size of the transition between the entrance window and thehighly reflective surface (the first reflection must be fully incident on the highreflector). Secondly, the lateral offset of the reflected beam must be greater thanthe width of the input beam plus the width of the coating transition. Normally,this condition is optimized when the beam waist is located where the inputbeam first reflects from the partial reflector (fig. 4.1).

To increase the extinction of elastically scattered light, that is orders of magnitudestronger than Brillouin signal, especially in presence of turbid media or optical re-flections, Ref. [45] proposed to cascade multiple VIPAs. The illustration in figure 4.2explains how the cascading can be beneficial. When a single VIPA is used the lightis dispersed in a single direction and if the unwanted elastic scattered light exceedsthe spectral extinction of the spectrometer, a crosstalk (dark green line in fig. 4.2a)appears on the spectral axis and the Brillouin signal could be overwhelmed by it.However it must be noted that, although the spurious signal is spatially overlappedto the Brillouin signal, it is spectrally separated; introducing a second VIPA, with itsdispersion axis orthogonal to the axis of the first VIPA, the “stray light” can be spa-tially separated from the signal: in particular, as shown in figure 4.2b, “stray light”is mainly confined to the horizontal and vertical axis while the signal is dispersedalong the diagonal. The introduction of an aperture mask between the two stages(fig. 4.2c) allows further spectral filtering of unwanted elastically scattered light.To implement the multi-stage configuration in a real spectrometer a lens is neededto focus the light onto the first VIPA; the angularly dispersed light is then focusedby a subsequent lens, spatially filtered with the mask and adjusted for the next stageby one or more lenses; after the last stage a lens images the final pattern onto thecamera. Two types of lenses are used: cylindrical lenses may be used when light mustbe focused/collimated only in one direction (either horizontal or vertical); sphericallenses instead operates on the two directions separetely (i.e. if light was divergent inthe horizontal direction while it was collimated in the vertical one, the spherical lenswill collimate it in the horizontal direction while focus it in the vertical one). Severalconfiguration of lenses are possible: a detailed description of the configuration usedin this work will be given in section 5.1.3.


Figure 4.2: Illustration of the principle of multiple VIPA cascading. (a) In a sin-gle stage, Brillouin signal and crosstalk overlap. (b) In a double spectrometer, thespectral axis is diagonal, while crosstalk is confined to horizontal and vertical direc-tions. (c) A mask between first and second stages can filter out part of the crosstalkcomponent. [45]

4.2.2 EM-CCDThe cross section of Brillouin scattering is very low, therefore when an optical powerof the order of mW for the pump laser is used3, the Brillouin scattered photonsthat reach the spectrometer are few thousands per second. To detect the Brillouinfrequency shift it is necessary to detect these few photons with a spatial resolution ofthe order of tens of micrometers. To achieve this goal an Electron-multiplying CCD(EM-CCD) was used that can detect up to a single photon while maintaining theresolution of a standard CCD.The sensor that detects the photons in a EM-CCD is basically the same as a stan-dard CCD: an array of quantum wells. When a photon is absorbed by the sensor, itgenerates a photoelectron that is trapped in the well. At this point it comes the firstelement in signal processing that is peculiar of EM-CCD: a storage area; the applica-tion of a series of voltages across the sensor forces the transfer of all photogeneratedelectrons from the imaging to the storage region of the detector. Doing so ensures theprocessing of the acquired image while performing a new acquisition. The electronsin the storage region are moved one by one into the multiplication register, using thesame technique as CCD redout process: electrons at the detector bottommost rowtravel pixel by pixel into the multiplication register. In the multiplication registerthere are several hundred electrodes. When a photoelectron encounters an electrode,there is a 1–2% probability that it will generate a secondary electron by impact ion-ization, a type of avalanche effect. As a result, an incoming signal of a few photonscan be amplified up to several thousand times. The charges then reach the outputamplifier where they are converted into an electric impulse, subsequently digitized toform an image. In figure 4.3 there is a schematic representation of the redout process.

3The impinging optical power can’t exceed few mW to avoid damaging the biological sample.


Figure 4.3: Schematic representation of the redout process in a EM-CCD. [49]

Listed below are the important parameters that determine the performances of anEM-CCD [49]:

thermal noise Due to thermal agitation the electrons present in the silicon substrateof the sensor can gain enough energy to become free and reach the quantumwells. Those electrons are indistinguishable from the photogenerated ones andare thus a source of noise. To reduce this phenomenon the sensor must be cooledand commercial EM-CCD are usually cooled up to -100°C.

redout noise When the charges are amplified and digitized, some noise is introducedin the process by the electronics. This noise, at some point, cannot be furtherreduced but if the signal is amplified before redout, the impact of the noise islower. The electron-multiplying register is responsible for amplifying the signalbefore the redout process.

clock induced charges To shift the photoelectrons in the signal reading and am-plification, a clock is applied that as high voltage and frequency. This processis an additional noise source since some electrons can be extracted from thesubstrate due to high voltage, especially at high frequency.

EM gain: The process of avalanche multiplication comes at the price of introducingthe EM register. This register is a sensitive component that may easily saturate,and saturation may lead to its premature aging or damage.

charge transfer efficiency When, in the reading process, the charges are trans-ferred from one potential well to the next, some photoelectrons may be left


behind increasing the signal in a pixel at the expense of the pixel nearby.

excess noise factor When a photoelectron enters the multiplication register thenumber of electrons produced by the amplification process is stochastic in na-ture. Such a process is described by a poissonian distribution that has a knownmean. Thus, especially when the number of photoelectrons being amplified islow, it is not possible to determine for sure their number from the number ofgenerated electrons.

4.3 Refractive index measurementAs shown in equation 4.1, the Brillouin shift Ω depends on the refractive index at theinteraction point n, among other parameters of the medium (sound velocity v) andof the scattering geometry (scattering angle θ). It is worth noting that θ is the anglebetween the incident and scattered light at the interaction point: after setting theincident and collecting angle outside the medium θ itself depends on the refractiveindex due to refraction at the surface air-medium. The problem can be simplified ifthe incident angle and the collecting angle are equal, as shown in figure 4.4. Theyare related to the scattering angle θ by the relation

θ = π − 2β = π − 2 arcsin(sinα

n

)In this case equation 4.1 becomes:

Ωα = 2vλn

√1− sin2 α

n2

To eliminate v dependence the shift can be measured at two different incident (=col-lecting) angles α1 and α2:

n =√

sin2 α1 − r2 sin2 α2

1− r2 , r = Ωα1

Ωα2

In particular if one of the two angles is 0°, corresponding to backscattering (θ=180°),the previous equation reduces to:

n = sinα√1−

(ΩαΩ0

)2(4.2)

To obtain equation 4.2, the assumption of uniform medium with refractive index nwas made. It can be shown however that it is still valid if the sample consists of manyparallel layers (fig. 4.5). In fact using the Snell law:

sin θk = nk−1

nksin θk−1 = nk−1

nk

nk−2

nk−1sin θk−2 = nk−2

nksin θk−2 = ... = n1

nksin θ1


Figure 4.4: Diagram of scattering geometry used for refractive index measurement.

that is the same refraction law that would be obtained at a virtual interface betweenthe last and the first layer. Moreover the condition of parallels layers must hold onlyalong the illuminated path, that is for a transversal size that is bigger but comparablewith the beam size.


1

2

3

4

k-1

k

2

n1

n2

n3

n4

nk-1

nk

Figure 4.5: Schematization of a sample made up by parallel layers.

Chapter 5

Experimental setup for ConfocalBrillouin Microscopy

In the previous chapter we underlined the dependence of Brillouin frequency shift onthe refractive index of the medium and presented a method for measuring it exploitingBrillouin scattering. In this chapter we will describe the setup built for a proof ofprinciple of the feasibility of such a measurement. Particular attention will be givento the factors that can potentially degrade the precision of the measurement.

5.1 Brillouin setupIn figure 5.1 it is shown a schematic of the setup used to measure the refractive indexexploiting Brillouin scattering. As it was pointed out in equation 4.2, the Brillouinfrequency shift must be measured in epidetection and in the angled geometry (shownin figure 4.4) in order to extract the refractive index. Thus there must be an efficientway to switch quickly1 between the two geometries without altering the alignment,which is highly critical for precision of the measurement. The solution adopted in thiswork is to use two different paths (and optics) for the two geometries and to selectthe desired geometry by rotating two ad hoc designed chopper disks simultaneously.We now describe more in details the setup, dividing the description into three logicalparts: path followed by pump laser, optics for signal collection, spectrometer.

5.1.1 Pump laser pathThe pump laser beam follows different paths in the two geometries. The beam origi-nating from the laser was split by means of a lateral displacement beamsplitter that

1For each point of 3D scan the signal from the two geometries must be acquired. A slow switchwould reduce substantially the speed of the overall scan.

5.1 Brillouin setup 45

λ/2

λ/4

SAMPLE

0

PBS

532nm SM

Laser 25 mW

DBS

VIPA

SPECTROMETER

OBJ_S

1

FL

FL

OBJ_C

OBJ_C SM 1x2

coupler

Pump (epidetection)

RAP

CD

CD

Pump (angled)

Signal (epidetection)

Signal (angled)

Figure 5.1: Setup for measuring Brillouin frequency shift at two different geometries.The selection of the geometry is done by simultaneous rotation of both chopper disks.DBS. lateral displacement beamsplitter, CD = chopper disk, PBS = polarizing beam-splitter, λ/4 = quarter waveplate, λ/2 = half waveplate, FL = 5-axes fiber launchersystem, OBJ_S = objective lens for focusing pump laser onto the sample and collectBrillouin light, OBJ_C = objective lens used for coupling Brillouin signal into thefiber, RAP = right angled prism.


produces two parallel beams at a fixed distance. The distance between the two beamsis then reduced to fit the half of the clear aperture of the objective lens by means ofa beam reducer (not shown in figure 5.1). Below is the description of the two beampaths:

epidetection the usual approach is used. The beam is centered on the objectivelens and focalized, along the optical axis, onto the sample. Part of the Brillouinsignal, that isotropically comes out from the interaction volume, follows back-wards the same optical path as the pump laser and is collected by the objectivelens.

incidence at an angle the incoming laser beam is not aimed at the center of theobjective lens but it is displaced at the limit of its clear aperture. In this way itwill be focused with an angle. As in epidetection, part of the Brillouin signal iscollected by the objective lens; to select only the part of the Brillouin scatteredphotons that comes from the right angle (that is the one equal to incident angle),a mirror was used for subsequent alignment.

To switch automatically between the two geometries, it was used an USB controlledstepper motor with a chopper disk for beam selection. The chopper disks was designedusing a 3D modelling software and was 3D printed in black plastic. In figure 5.2 it isshown the principle of beam blocking: the two parallel beams reach the chopper disk;based on the position of the disk, one of the two beam is blocked while the other canpass through.The beams reach microscope objective, which is mounted vertically on a microscopeassembly, by means of a 45° mirror that directs the light upwards.

5.1.2 Signal collectionThe Brillouin signal is generated in the focal volume. In epidetection the excitationand collection volumes coincide. This is not true in the angled geometry, in whichthe intersection volume is reduced, as shown in figure 5.3. If, for the sake of makingan estimation, the focal volume is considered to be a parallelepiped and the angle θbetween the direction of incidence and collection is 90°, the overlap of the collectionand excitation volume is:

%overlap = VoverlapVtot

= (2√

2w0)3

(2√

2w0)2b= 2√

2w0

b

therefore, using the expression for b = 2πw20

λ:

%overlap =√

2λπw0

(5.1)


Figure 5.2: 3D model of the chopper used for beam selection in the setup shown infigure 5.1.

θ

Figure 5.3: Overlap (blue region) of two gaussian beams that intersect at an angle θ.


Figure 5.4: Detail of the setup used to separate the two Brillouin signals and couplethem into single mode fibers. On the sides of the right angled prisms there are thetwo 5-axes fiber launcher systems.

Assuming an uniform intensity in the focal volume, the percentage of overlap is equiv-alent to the efficiency of collection.A fundamental step in signal collection is the rejection of specular reflection, whichcould easily exceed the extinction ratio of the spectrometer if it is not reduced. Polar-ization of the laser is exploited at this end: the pump laser, that is linearly polarized,passes through a quarter waveplate oriented at 45° with respect to polarization di-rection of the laser. In this way its polarization becomes circular. When light isreflected back by the sample its handedness is reversed and, when it passes thoroughthe quarter waveplate again, its polarization is made linear but in a direction that isperpendicular to the incident one. Therefore when reflected light reaches the polariz-ing beam splitter is ideally totaly transmitted while the Brillouin signal has randompolarization thus it is half reflected and half transmitted. By taking only the reflectedpart of the light the specular reflection is removed.The signal from the two geometries needs to be coupled into a single mode fiber tobe sent to the spectrometer. At this end two microscope objectives were used thatfocus the light onto the tip of a fiber mounted on a 5-axes fiber launcher system,that allows fine alignment to maximize coupling efficiency. Since the Brillouin beamscoming from the two geometries are parallels, after the objective lens, with a distanceof about 1 cm, there was not enough space for placing the two launcher system sideby side. Thus a right angled prism was used to separate the two beams and sendthem in opposite directions (fig. 5.4).The selection of the signal, by means of the chopper disk, is also necessary in collec-


VIPA

V1

Cylindrical

lens CL2 Slit

Lens

L1

VIPA

V2

Lens

L2

Achromatic

doublet pair

ADP Cylindrical

lens CL1

EM-CCD

Fiber

collimator

Figure 5.5: Schematic of the spectrometer used for Brillouin shift measurement inthis work.

tion. This is because Brillouin scattering is isotropic. Thus, if for example the pumplaser is in epidetection configuration, the signal would come from both epidetectionand angled geometry if no selection is done.

5.1.3 SpectrometerLight to be analyzed is sent to the spectrometer through a single mode optical fiber.The first step is to collimate the light coming out from the fiber by means of a fibercollimator. Then a series of lenses and VIPAs produces a pattern, that encodes thespectrum of the light. This pattern is finally imaged by the EM-CCD. All the lenseshave a focal length of 200 mm while the achromatic doublet pair is a 1:1 pair. BothVIPAs have a FSR of 20 GHz. The EM-CCD is an Andor iXon cooled up to -80°.We now describe in details the role of each optical element in the spectrometer (fig.5.5).

CL1 The first cylindrical lens CL1 focuses the light in the horizontal direction whilein the vertical direction it is left unchanged: a vertical line is formed in the focalplane of CL1. The choice of its focal length determines the focusing angle that,as explained in section 4.2.1, must be as small as possible; therefore a long focallength (200 mm) is used.

V1 The back surface of the VIPA V1 is in the focal plane of CL1, so that a verticalline is presented at his input. V1 disperses the light in the horizontal directionwhile in the vertical direction it remains unchanged.

5.2 Spectrometer calibration 50

CL2 A second cylindrical lens CL2 converts the angular dispersion into a spatialdispersion producing the image of the VIPA pattern onto a subsequent slit, thatis in the focal plane of CL2. In the vertical direction the light is still collimated:the first stage of the VIPA operates only in the horizontal direction.

slit The slit is used to select the order of the VIPA and cut the unwanted laser light.The alignment of the slit is critical since, if it is not correctly imaged on thecamera, the light diffracted by its edges can invade the area where the Brillouinsignal is present, thus reducing the extinction of the spectrometer.

L1 After the slit the light is angularly dispersed in the horizontal direction while itis collimated in the vertical direction. The lens L1 collimate the light in thehorizontal direction while focusing it in the vertical direction, in order to sendit to the second stage that operates in vertical direction.

V2 The second VIPA V2 is in the front focal plane of L1 and it disperses the light inthe vertical direction while leaving it collimated in the horizontal direction.

L2 The lens L2 converts the angular dispersion into a spatial dispersion and focalizesthe light in the horizontal direction. In the front focal plane of L2 the finalpattern is formed, that has the Brillouin signal on the diagonal.

ADP The doublet pair ADP is used to image the pattern onto the camera sensor.

5.2 Spectrometer calibrationThe calibration of the spectrometer is critical, since to measure small variations onrefractive index (few percent) high accuracy is needed for the ratio of Brillouin fre-quency shift between the two geometries. Biological samples are mainly composedof proteins in water solution, thus they exhibit refractive index close to 1.33 withvariations that are typically on the second decimal digit and these variations must bemeasured in order to obtain useful information.Let’s calculate how the error on the frequency measurement affects the value of therefractive index, in the assumption that the error on frequency is statistical2. Re-calling the expression 4.2, if the ratio r = Ωα

Ω0between the Brillouin frequency shifts

measured in the two geometries (Ωα and Ω0) has an uncertainty σr, the uncertaintyon the refractive index σn will be:

σnn

= n∗√n∗2 − 1 · σr, n∗ = n

sinα2Note that the error is likely to be systematic, in case it is due to calibration.


mean

a b

c

Figure 5.6: (a) raw spectrum as acquired from the camera. (b) the image is rotatedby a suitable angle to make the spectral axis horizontal. (c) a few pixels (markedby the red arrow in panel b) are averaged to obtain a one-dimensional spectrum; alorentzian fit is performed to obtain the position of the peaks.

Therefore the relative error on the refractive index n is approximately equal3 to theabsolute error on the ratio r between the Brillouin frequency shifts measured in thetwo geometries. Since r . 1 to obtain a relative error on n of few percents the relativeerror on r must be even smaller.Note also that the error on n is proportional to n∗. Thus the smallest the angle αthe larger the error on n. Therefore α is chosen as large as it is allowed by the NA ofthe objective lens.Before describing the approach used to calibrate the spectrometer, we first presentthe method used to determine the position in pixels of the peaks from the 2D imagesacquired by the camera (fig. 5.6). Since the spectral axis is on the diagonal, the imagemust first be rotated; the rotation angle is chosen in the calibration process and itis the same for all the subsequent measurements. Few lines of pixels, containing thesignal, are then averaged to obtain a one-dimensional spectrum; a lorentzian fit isperformed to determine the position of the peaks. The two peaks shown in figure 5.6are the stokes (on the right) and antistokes (on the left) peaks. The laser is not shownin the figure but would consist of two peaks, one on the far right and the other on thefar left. The reason behind the presence of a double peak of the laser is that everyFSR the same pattern is repeated (with different distances due to different dispersive

3n∗√n∗2 − 1 ≈ 1


power at higher orders). The periodicity of the pattern is also the reason why theantistokes peak is on the left of the stokes peak, although it has higher frequency.Now the calibration method is described. Let’s call p1 the distance in pixels betweenthe first laser peak and the Stokes peak and p2 the distance between the antistokespeak and the second laser peak; p1 and p2 are obviously related to the Brillouinfrequency shift in GHz. If the dispersion of the VIPA was linear and there weren’toptical abberations, p1 would be equal to p2 (stokes and antistokes have the samefrequency displacement from the laser frequency). However experimentally p1 6= p2and that means that dispersion is not linear and more than two points are needed forcalibration. Liquids with known Brillouin frequency shift were used (water, ethanol,methanol) to calibrate the spectrometer.The first approach that was used for calibration is to infer the Brillouin frequency shiftfrom the values of p1 and p2. This requires that in every measurement the positionof the laser must be known. Especially in epidetection, laser light is too intense withrespect to Brillouin signal and measuring both signals at the same time would saturatethe camera. Therefore laser light needs to be cut with a slit and it is not possible tomeasure p1 and p2 from a single frame containing only Brillouin signal. A possiblesolution would be to measure the laser position once as first measurement and touse its position for all subsequent measurements; unfortunately laser frequency drifts(and sometimes jumps) on a time scale of few minutes. Therefore for an accuratemeasurement the laser position must be determined at every acquisition.A second possibility for calibration is to relate the distance in pixels between the twopeaks dp = p2− p1 to the Brillouin frequency shift. In appendix B it is demonstratedthat this relation exists if the Brillouin frequency shift is lower than the FSR andthe relation between the calibration coefficients that are found using this method andthe previous one is calculated. The main advantage of this approach is that only thedistance between the Brillouin peaks is needed to calculate the frequency shift, so itshould be resilient to laser fluctuations4 without the need of acquiring frequently theposition of laser peaks. A drawback is that at every calibration material it correspondsonly one point in the calibration curve, while in the first approach the points are two(stokes and antistokes): this is a limiting factor due to lack of Brillouin frequencyshift of materials in literature. Another limit is that this approach cannot be usedif the Brillouin frequency shift is higher than the FSR, but this is not a problem forbiological samples that present Brillouin frequency shift very close to that of water.In figure is shown the calibration curve. Only two materials and the laser were used forcalibration5 because data of Brillouin frequency shifts of liquids is scarce in literature

4The nonlinearity of dispersion of the VIPA could cause a variations of calibration coefficientswhen the laser drifts but, since the frequency drift corresponds to less than a pixel, this effect shouldbe negligible.

5At first ethanol was also used but it was then removed from the calibration curve due to con-tamination (ethanol dissolved the dish).


y = 0.0002x2 - 0.1296x + 10.353

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0 20 40 60 80 100

Tab

ula

ted

Bri

llou

in S

hif

t (G

Hz)

Peaks distance (px)

Calibration curve

water

methanol

laser

Figure 5.7: Calibration curve for the spectrometer: a second order polynomial is usedin fitting (shift(GHz) = a · dp2 + b · dp + FSR

2 ).

and sometimes the conditions of measurements are either different from ours or notspecified (Brillouin shift is dependent on temperature). Due to the lack of calibrationmaterial a second order polynomial is fitted to the data:

shift(GHz) = a · dp2 + b · dp + FSR

2In figure 5.7 the calibration curve is shown. The FSR is found to be 20.7 GHz thatis very close to value of both VIPAs (20GHz) as declared by the manufacturer.

5.3 Experimental dataFor the proof of principle some uniform liquids were used as a sample. The liquidunder analysis was poured into a plastic Petri dish. The dish had a window of 200µm thickness, ideal for using with short working distance objective lenses (0.5 mm inour case).

5.3.1 Data acquisition and analysisUsing the Andor Solis software, 20 frames were acquired for each geometry in a kineticacquisition. By means of a Matlab script each single frame was automatically rotated,


10 20 30 40 50 60 70 80 90 100pixels

0

0.2

0.4

0.6

0.8

1

inte

nsity

(a.

u.)

Brillouin signal for methanol at two geometries

epi

angle

Figure 5.8: Brillouin frequency shift in the two different geometries for methanol.

averaged and fitted with a double Lorentzian curve, as previously explained and shownin figure 5.6. In figure 5.8 it is shown the Brillouin frequency shift for methanol inthe two geometries as an example. The difference in the Brillouin frequency shiftbetween the two geometries is clearly visible. Notice that the spectrum from theangled geometry is more noisy, due to lower signal: to have a comparable number ofphoton counts from the two geometries, the exposure time was 20 times longer in theangled geometry (2s) than in epidetection (0.1s). As explained before, in the angledgeometry the signal is expected to be lower due to reduced overlap between pumpingand collecting focal volumes. Using equation 5.1 with w0 ≈ 3µm the overlap is foundto be about 8%. Since equation 5.1 is just an estimation, this phenomenon alonecould account for the full loss of signal of about 95% between epidetection and angledgeometry.Afterward the values of Brillouin frequency shift for each frames were averaged andthe standard deviation was calculated. The standard deviation of the mean was verylow, less than 0.1%, but it is not a good estimate of the error. In fact it doesn’ttake into account the error on the calibration parameters and actually this should bethe main source of error. Unfortunately there are not enough calibration points toestimate the error on calibration parameters.

5.3.2 ResultsGiven the Brillouin frequency shift in the two geometries, Ωα and Ω0, the functiony = 1√

1−(

ΩαΩ0

)2was calculated and plotted against the refractive index measured with

a refractrometer (figure 5.9). The refractometer has a resolution on the fourth decimal


DMSO 50%

Methanol

N2

Sucrose solution

Water

y = 1.217x + 0.383

1.98

2.00

2.02

2.04

2.06

2.08

2.10

2.12

2.14

1.32 1.34 1.36 1.38 1.40 1.42 1.44

(1-r

2)-1

/2

RI from refractometer

Figure 5.9: Plot of y = 1√1−(

ΩαΩ0

)2against the refractive index measured with a

refractometer. The expected relation is linear with a zero intercept and a slope thatequals the inverse of the sine of incidence angle α in the angled geometry, as describedin eq. 4.2.

digit and repeated measurement showed only small fluctuations on the last digit.Recalling eq. 4.2, the expected relation between y and n is linear with a zero interceptand a slope that equals the inverse of the sine of incidence angle α in the angledgeometry. Despite the number of data points being limited, data show a linear trendwith a correlation coefficient R2 = 0.995; the χ2 was not calculated since we don’thave a good estimate of errors due to calibration. Performing a linear fit on data theslope was found to be 1.217, that corresponds to an angle α of about 55°, and theintercept was found to be 0.383.

5.3.3 DiscussionTo be consistent with relation 4.2 the intercept must be null and the slope must behigher (smaller angle), since, based on the numerical aperture of the objective lens,


the angle should be of about 45°.This discrepancy can be attributed to several reasons.

• In epidetection the angle of incidence was assumed to be exactly zero. If thereexists a small tilt of the Petri dish with respect to the incoming beam theprevious condition is no more true but equation 4.2 is still valid in case of perfectalignment. In fact the assumption behind epidetection is that the angle betweenincoming beam and scattered light is zero and that assumption is true if lightis focused and collected through the same path. However, due to misalignment,there may exists a difference between the two paths; the difference impliesan angle between pump and scattered light. If we approximately model thiscircumstance using eq. 4.3, in the assumption of small α2 and maintaining theprevious definition of y, we obtain:

y = n√sin2 α1 − sin2 α2

√1− sin2 α2

n2 ≈ n

sinα1

(1− sin2 α2

2n2

)

This model could account for a non-null intercept. However the fit of thismodel to the data is worst than the linear model, thus it may be concluded thatmisalignment in epidetection is not the main source of error or the model is notappropriate.

• If, in the angled geometry, the incidence and collection angle are different equa-tion 4.2 is not valid and a new model would be needed. In this condition itis not easy to build an exact model: that is the reason why the “symmetric”geometry was chosen. Therefore much effort should be made in making surethat the “symmetry” condition is fulfilled. For this purpose the alignment ofthe coupling fiber was done with a mirror as a sample, so that reflection angleis equal to incidence angle.

• Equation 4.2 was derived in the assumption of delta-like incidence and collectingangles. The real situation is pictured in figure 5.10. If a gaussian distributionis assumed for the light impinging on the focusing objective lens

I(x) = I0e− (x−x0)2

2w2

then, in the limit of geometrical optics6, the angular distribution of the lightafter the lens is:

I(α) = I0

cos2 αe−

f2(tanα−tanα0)2

2w2 = I0

cos2 αe− f2 sin2(α−α0)

2w2(cosα cosα0)2

6Since we are interested in the angular distribution in the focal volume, where the signal comesfrom, the geometrical optics approach may not be valid. Anyway we won’t use this results forcalculation but just to make some considerations.


O

f

xx0

2w

α0

Figure 5.10: Focusing of a beam that is not on the optical axis. Light is focused witha distribution of angles α.

that, for small δα = α− α0, becomes:

I(α) ≈ I0

cos2 α0[1 + (2 tanα0)δα]e−

f2δα2

2w2 cos4 α0

Note that the distribution of δα is the product of [1 + (2 tanα0)δα], that makesit not symmetrical, and a gaussian having a standard deviation σα = w cos2 α0

f.

The same calculation is true for the angle of collection. The finite width ofangular distribution can have two effects: enlarge the width of the Brillouinfrequency spectrum and change its shape and position. In our case the en-largement wouldn’t be a problem but if the shape and position are altered anerror in the ratio Ωα1

Ωα2may be introduced: this would introduce an error on both

the slope and the intercept. It is not easy to quantify this error but for smallbeam size it should be negligible, since the width of angular distribution σα isproportional to the beam size before the objective lens.

• The error σr on ratio r between the Brillouin frequency shift in the two geome-tries, that may be introduced by non optimal calibration of the spectrometer,is another source of error. Let’s estimate its relevance making the assumption


that it can be treated as a statistical error. Using the previous definition of y:

σy = dy

drσr = r√

(1− r2)3σr

that, written in terms of n∗ = nsinα , is:

σy = n∗2√n∗2 − 1 · σr

In general, if a least squares fit (y = mx+ q) is performed on experimental dataaffected by an error σy, the slope m and the intercept q have an error [50]:

σq =

√√√√ x2

x2 − x2 ·σy√N

σm =√

1x2 − x2 ·

σy√N

where N is the number of data points and is the average.In our case the independent variable is x = n while the dependent variable isy = 1√

1−r2 . Using the data shown in figure 5.9, it is found σy ≈ 7.6σr, x = 1.377,x2 = 1.8978 and N = 5. Thus:

σq =√

1.897791.89779− 1.3772 ·

7.6σr√5≈ 115σr

σm =√

11.89779− 1.3772 ·

7.6σr√5≈ 83σr

Therefore an error of less that 1% on r can fully account for the discrepancybetween the calculated and the expected parameters.

From the previous discussion it is clear that the approach of measuring refractiveindex exploiting the frequency dependence of Brillouin scattering on the refractiveindex in the interaction volume is very sensitive to the error on the measured Brillouinfrequency shift. Therefore the first step in improving the method is the definition ofa calibration method that is both very accurate and constant in time (i.e. that is notinfluenced by the fluctuation of laser frequency or other mechanical vibrations of thespectrometer).

Conclusions and future work

The aim of this work was to introduce Confocal Brillouin Microscopy as a techniquefor measuring 3D maps of refractive index and to compare it with a widely usedtechnique, Digital Holographic Microscopy (DHM).DHM is a technique for quantitative phase imaging. It exists in many configurationsand a lot of effort has been made, both in the theoretical and experimental field, inorder to obtain best performances in term of resolution, accuracy, easiness of opticalsetup, effectiveness of reconstruction algorithms, aberration-free images. Indeed thiseffort is justified by the benefits that this technique offers:

• the accuracy is very high: phase noise of less that λ/1000 has been reported.

• it is very fast: in a single acquisition (usually of tens of microseconds) a whole2D phase image can be obtained.

• it requires a very simple optical setup: just few lenses and beam splitters and,in same configurations, even none of them.

• reconstruction algorithms are very efficient: they make use of tools like Fouriertransform that can be computed quickly with low computational cost.

• real time imaging of living biological samples is possible.

On the other end, referring in particular to refractive index measurement, there aresome limits:

• phase wrapping is unavoidable when the difference between optical path insidethe sample and the medium is larger than a wavelength; phase wrapping isactually a loss of information: to recover the phase correctly either opticalmethod must be used or some assumptions on the phase must be made toapply the appropriate unwrapping algorithm.

• the thickness of the sample must be known in order to retrieve integral refractiveindex from phase information; either some assumptions must be made on theshape of the sample or other techniques must be used to measure it.


• only relative refractive index can be measured and the spatial scale of refractiveindex variation must be comparable with the resolution; otherwise phase jumpsare invisible to unwrapping algorithms and that causes a failure of correct phasereconstruction. In order to better explain the problem let’s consider the extremecase of a sample made up by two plates of same thickness but different refractiveindex, placed side by side, so that the optical path of the first plate is exactlyone wavelength larger than the other: light that travelled through both part ofthe sample has exactly the same wrapped phase and there is no way to tell thedifference in refractive index between the two.

• reconstruction of 3D map of refractive index requires rotation of the sample orof the beam

Refractive index measurement exploiting Brillouin scattering should be able to over-come those difficulties. The idea of collecting Brillouin scattered light from differentgeometries to measure refractive index dates back to the eighties [46]. Anyway, as itwas pointed out, acquisition time was up to several hours and it was unimaginableto obtain a 3-dimensional map of a sample. Only in the last decades, thanks to theintroduction of VIPA spectrometer, it has been possible to implement scanning tech-niques. With respect to DHM techniques, Brillouin scattering for refractive indexmeasurement has some disadvantages:

• higher cost, mainly due to the very narrow linewidth of laser and the EM-CCD

• higher complexity in the optical setup, especially in the spectrometer part

• slower, due to its confocal nature that requires scanning

On the other hand it would have the great advantage of an absolute and intrinsically3D-resolved measurement of refractive index. This is a great improvement inthe analysis of biological samples.An example of the importance of 3D map of refractive index is cells nuclei: thereis a controversy whether the nucleus has a higher or a lower refractive index withrespect to the cytoplasm. Common sense would suggest that, since nucleus is apacked object, it has higher refractive index. However there are controversial reportsabout this question. A 2016 work [34] measured refractive index of nuclei isolatedfrom four different cell lines. For the measurement off-axis DHM was used. Theextraction of nuclei from the cells was necessary since measuring nuclear refractiveindex within a cell is complicated by its convolution with nuclear height and thecontribution of the surrounding cytoplasm and organelles. They also showed thatthe extraction of the nuclei shouldn’t alter the result. The result was the same forall the cells: nuclei have lower refractive index compared to that of the whole cells.Conclusion of Ref. [34] gives an insight on the importance of this result: “a lower


nuclear RI compared to the whole cell has not only implications for technologicalapplications where the RI distribution within biological samples plays a crucial role,but it also changes our perception of the nucleus in light of its biological functions.Quantifying physical properties of biological material thus continues to contribute toa better understanding of the intricacies of living matter.”Using Brillouin it would be possible to do the measurement in situ without altering thecell. From the previous example it should be clear that measurement of a 3D map ofrefractive index in biological samples can provide important information about basicmorphology.Although the sample used in this work was a uniform liquid and no 3D map wasmeasured, it was demonstrated that Brillouin confocal microscopy can be extendedfor refractive index measurement. Since Brillouin confocal microscopy is capable of3D mapping, also the proposed technique should allow 3D-resolved refractive indexmeasurement. In order to verify experimentally this statement, some steps must betaken:

• calibration of the spectrometer must be improved, in order to have higher ac-curacy on refractive index measurement.

• resolution must be improved to be good enough for imaging of biological sam-ples; the axial resolution of the actual setup is about a hundred of microns thusit is not usable for cell measurements.

Appendix A

Branches cut unwrappingalgorithms

The theory of unwrapping algorithms is from Ref. [51], a book published in 1998that comprehensively analyzes the problem of 2D unwrapping and presents all majoralgorithms.

A.1 Phase residuesAs mentioned in section 2.4, if the wrapped phase is not “ideal” and it is unwrappedalong a one-dimensional path, the result will depend on the choice of the path. Theproblem arises because of residues. A residue is a singularity, inside a “loop” of fourpixels, for which the integral of the phase derivatives (i.e. the sum of the wrappedphase differences) is not zero, but 2π or −2π. Based on the sign of the integral1the residue is said to be positive or negative. If there are no residues the phase canbe unwrapped along any path and the result will be the same. In case residues arepresent the path must be chosen so that it encloses an equal number of positive andnegative residues or none residues at all; under this condition there are theoremswhich assure that the result of the unwrapping will be independent of the choice ofthe path.In figure A.1a there is an example of an image with a residue (gray square at thecenter) and three possible paths that go from point A to point B are drawn . If oneof the two lateral paths is chosen, the result of unwrapping will be approximately6π because three phase jumps are intercepted; if the central path is chosen instead,there will be only two phase jumps thus the phase will be 4π. A way to solve thisproblem is to impose a branch cut that restricts the possible paths connecting A toB (fig. A.1b), so that positive and negative residues are balanced or a path cannot

1By convention the sum is calculated in a counterclockwise direction.

A.2 Goldstein’s branch cut algorithm 63

Figure A.1: Path-dependent phase unwrapping in the presence of residues. (a) Phasedata with a positive (+) and a negative (-) residue. The unwrapped value at thepoint B depends on the path followed from the point A. The straight and the curvedpaths yield different values. (b) If a branch cut connects the residues, the unwrappedphase is the same for all allowed paths. (c) The branch cuts may also connect theresidues to the border. In this case the phase is unwrapped differently. [51]

turn around any residue. Of course the choice is not univocal (fig. A.1c) and the finalresult will depend on the disposition of branch cuts. It is thus necessary to establishsome criteria to guide their placement. An essential requirement is that branch cutsmust not completely isolate areas of the image, whose phase would become unrelatedwith the rest of the image. A strategy could be to choice the shortest cuts possible;in some applications, this choice is statistically demonstrated to be the best strategy.

A.2 Goldstein’s branch cut algorithmBasically three steps are made in path-following unwrapping algorithms that use thestrategy of residue compensation:

• identification of residues

• generation of branch cuts

• path integration around the branch cuts

The algorithm by Goldstein, Zebker and Werner is a fast algorithm that can findan approximate solution to the problem of determining the optimal configuration ofbrunch cuts (i.e. the cuts whose sum of lengths is the shortest possible) that balancesall the residues. This algorithm allows the brunch cuts to also link “clumps” of (i.e.more than two) residues together.

A.2 Goldstein’s branch cut algorithm 64

After the algorithm detects the first residue2, a 3x3 box is centered around it (i.e onepixel around the residue is taken). If another residue is found inside the box, regardlessof its polarity, it is connected with the first one. If the second residue has oppositepolarity compared to the first one, the residues are labeled as “balanced”. In the othercase the algorithm continues the search for residues in the 3x3 box and, if no otherresidue is found, the box is first centered to all the residues connected together then,if residues are still unbalanced, the box is extended to 5x5 and the same operationis repeated. The search ends when the sum of polarities of the residues connectedtogether is zero. If the border of the image is reached before this condition is met, abrunch cut is used to connect the clump of residues to the border.Once a clump of residues is balanced, the algorithm searches for other unbalancedresidues and the same steps are repeated.

2A residue is identified by 4 pixels but it is treated by the algorithm as a single pixel (for examplethe upper left pixel of the 2x2 square).

Appendix B

Spectrometer calibration

Let p be the first peak position in pixels and ν the corresponding Brillouin frequencyshift. Given the calibration function g(p) : p → ν, that at every pixel assigns itsfrequency in GHz in the FSR domain, we know that, in the assumption that theBrillouin peaks we observe are not switched (i.e. Brillouin frequency shift is less thanFSR/2), g(p) must be a bijection; thus, g−1 exists.Let’s define the function γ(p) : p → dp that at every pixel assigns the distance inpixel between the two Brillouin peaks. Using the function g we can write: γ(p) : dp =g−1(FSR− g(p))− p, where p is the position of the first peak in pixels.It is clear that γ is a monotonically decreasing function, since the higher the Brillouinshift, the lower the distance between the two peaks. Therefore, γ is also a bijection,and we can write: γ−1(dp) : dp → p. In the given domain the two functions can becomposed in a third one h(dp) = g γ−1 : dp → p→ s, which is the function used forthe calibration.Let’s assume g has a second order polynomial shape:

g(p) = ap2 + bp

The inverse function will be

g−1(ν) : p = −b±√b2 + 4aν

2awe can now write γ

γ(p) : dp = g−1(FSR− g(p))− p =−b±

√b2 + 4a(FSR− ap2 − bp)

2a − p

Inverting the previous equation for dp

2a(dp + p) + b = ±√b2 + 4a(FSR− ap2 − bp)

66

4a2(dp + p)2 + b2 + 4ab(dp + p) = b2 + 4a(FSR− ap2 − bp)a(dp + p)2 + b(dp + p) = FSR− ap2 − bp

Observing that dp + p is the position, in pixels, of the second peak and (ap2 + bp) isthe Brillouin frequency shift, we note that the latter equation expresses the symmetryof the two peaks in the frequency domain; extracting p from it:

γ−1 : p =−(adp + b)±

√(adp + b)2 − 2a(adp2 + bdp − FSR)

2a

Composing the two function (using Mathematica)

h = g γ−1 : s = ap2 + bp = FSR

2 ± dp2

√2aFSR− a2dp

2 + b2

Where the± gives the two switched Brillouin shift corresponding to the same distance.In our case, we will only consider samples with a shift lower than FSR

2 .

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3D mapping of refractive index exploiting Brillouin scattering

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