Measurement of all orthogonal components of displacement in the volume of scattering materials using...

Measurement of all orthogonal componentsof displacement in the volume of scattering

materials using wavelengthscanning interferometry

Semanti Chakraborty and Pablo D. Ruiz*

Wolfson School of Mechanical & Manufacturing Engineering, Loughborough University, Loughborough LE11 3TU, UK*Corresponding author: [email protected]

Received May 29, 2012; revised July 4, 2012; accepted July 5, 2012;posted July 5, 2012 (Doc. ID 169564); published August 7, 2012

A wavelength scanning interferometry system is proposed that provides displacement fields inside the volume ofsemitransparent scattering materials with high spatial resolution and three-dimensional (3D) displacement sen-sitivity. This effectively extends digital speckle pattern interferometry into three dimensions. The sample is illu-minated by three noncoplanar collimated beams around the observation direction. Sequences of two-dimensionalinterferograms are recorded while tuning the laser frequency at a constant rate. Different optical paths along eachillumination direction ensure that the signals corresponding to each sensitivity vector do not overlap in the fre-quency domain. All the information required to reconstruct the location and the 3D displacement vector of scat-tering points within the material is thus recorded simultaneously. A controlled validation experiment isperformed, which confirms the ability of the technique to provide 3D displacement distributions inside semitran-sparent scattering materials. © 2012 Optical Society of America

OCIS codes: 110.6955, 110.4500, 120.3180, 120.6160, 120.4290, 120.5050.

1. INTRODUCTIONExperimental mechanics is currently contemplating tremen-dous opportunities of further advancements thanks to a com-bination of powerful computational techniques and alsofull-field noncontact methods to measure displacement andstrain fields in a wide variety of materials. Identification tech-niques, aimed to evaluate material mechanical properties gi-ven known loads and measured displacement or strain fields,are bound to benefit from increased data availability (both indensity and dimensionality) and efficient inversion methods,such as finite-element updating and the virtual fields method(VFM) [1–3]. They work at their best when provided withdense and multicomponent experimental displacement (orstrain) data, i.e., when all orthogonal components of displace-ments (or all components of the strain tensor) are known atpoints closely spaced “within” the volume of the materialunder study. Although a very challenging requirement, an in-creasing number of techniques are emerging to providesuch data.

Techniques such as neutron diffraction and x-ray diffrac-tion can provide 3D strain fields in crystalline materials effec-tively measuring changes in the lattice parameter, but requirelong acquisition times for each measured point and rely onspatial scanning [4,5]. Digital volume correlation (DVC)[6,7] has gained popularity over the last few years for volu-metric strain and displacement measurements. It effectivelyextends digital image correlation (DIC) and particle imagevelocimetry [8], used from the early 1980s to measure surfacedisplacements and fluid flow. DVC has been used on datavolumes acquired with a diverse range of techniques to eval-uate 3D deformation fields in a variety of materials, and some

examples include: x-ray computed tomography (CT) of bones,sugar packs and silicon rubber phantoms [9–11]; scatteredlight imaging using sheet illumination in polymer phantoms[12]; ultrasound imaging of breast tumors [13]; confocal micro-scopic imaging of agarose gel phantoms [14]; optical coher-ence tomography (OCT) (also known as OCT elastographywhen combined with DIC or DVC) using two-dimensional(2D) cross-correlation speckle tracking to compute deforma-tion of gelatine scattering models, muscle tissue, and skin [15].In all these cases, spatial features that are part of the materialmicrostructure are tracked to subpixel accuracy. Magnetic re-sonance imaging (MRI) has also been used to evaluate 3Ddeformation inside bones and tendons [16,17] in conjunctionwith 2D DIC. An alternative method based on high-resolution3DMRI volumes and a surface registration algorithm based onfinite element optimization was used to evaluate the growth ofcortical surfaces during the folding of cerebral cortexof human and animal brains [18]. Also, 3D bulk displacementsmeasured by stimulated echo MRI were used in conjunctionwith VFM to reconstruct the stiffness ratio between aninclusion and its surrounding medium in a phantom [19].

Tomographic photoelasticity relies on the retardation be-tween different polarization components as light propagatesthrough a transparent material that shows transient birefrin-gence when loaded. It requires careful experimental measure-ments and inversion algorithms to retrieve all the componentsof the stress tensor [20]. Polarization sensitive OCT (PSOCT)also relies on the photoelastic effect to measure strains but italso provides depth-resolved distributions of birefringence,optical axis orientation, and phase retardation due to tissuefiber orientation [21]. It is effectively a type of reflection mode

1776 J. Opt. Soc. Am. A / Vol. 29, No. 9 / September 2012 S. Chakraborty and P. Ruiz

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photoelasticity that uses scattered light, with the ability to im-age structures and features within the volume of scatteringmaterials. It provides the difference between the in-planeprincipal strain components.

A family of optical techniques has recently emerged, all ofwhich can be considered as a type of phase-contrast OCT [22],also related to Doppler OCT [23]. They have the ability to mea-sure displacement fields inside the volume of scattering ma-terials without requiring birefringence, as opposed to PSOCTor tomographic photoelasticity, and with interferometric sen-sitivity. These techniques include: (1) Wavelength scanninginterferometry (WSI), which relies on a tunable laser. WSI sys-tems with out-of-plane sensitivity were proposed to measuredepth-resolved displacements of a stack of optically smoothglass–air interfaces [24] and also of scattering surfaces [25];(2) Tilt scanning interferometry (TSI), in which a temporal se-quence of monochromatic speckle interferograms is recordedwhile the illumination beam is tilted at a constant rate, wasshown to provide 3D displacement fields with one in-planeand out-of-plane sensitivity within an epoxy beam [26];(3) Phase-contrast spectral OCT (PC-SOCT) is based on abroadband source and spectral detection. A system was pro-posed to measure out-of-plane displacements in cross sec-tions of porcine corneas due to changes in intraocularpressure [27] and also one in-plane and the out-of-plane com-ponents in cross sections of a polymer phantom [28]. Noticethat in neither of these instances were all the components ofthe displacement field measured within the material volume.It is interesting to note that WSI, TSI, and PC-SOCT can all bedescribed in a common mathematical framework, as linearfilters in 3D space [29].

In this paper, we describe a method to measure a 3D dis-placement field within a semitransparent scattering materialusing WSI. All the components of the displacement vectorare obtained in each voxel of the data volume. This is achievedby simultaneously measuring the interference signal produced

for different illumination directions during a wavelength scan.These are separated in the frequency domain by adjusting theoptical path difference between the illumination and refer-ence beams, thus introducing different carrier frequenciesto the interference signal for different illumination directions.In Section 2, the optical setup is described. In Section 3, wedescribe the principle of WSI, the new approach based onfrequency multiplexing, and the main elements of themathematical framework for WSI. In Section 4, data acquiredfor a reference opaque surface is used to evaluate thesensitivity vectors of the interferometer, rather than establish-ing them from approximate illumination and observationdirections measured from the setup. The relationship betweenmeasured phase and calculated displacements is thus estab-lished in a convenient reference system. In Section 5, wepresent experimental results of the 3D distribution of thedisplacement field within a test sample that has undergoneknown rigid body in-plane rotation and out-of-plane tilt. Allthe components of the displacement vector at each voxel ofthe data volume are measured and compared to the nominalvalues of rotation and tilt introduced in the experiment viatwo rotation and tilt stages. Finally, in Section 6 we presenta discussion on potential applications, sources of error, andopportunities for further refinement, and we derive someconclusions.

2. OPTICAL SETUPFigure 1 shows a top view of the optical setup. The lightsource consists of a tunable laser, TL (TSL-510 Type A, SantecLtd.), which can scan the wavelength from 1260 to1360 nm insteps as small as 0.011 nm. This corresponds to a frequencyscan from 237.9305 to 220.4356 THz with a 0.002 THzfrequency step. The laser is connected via a single-mode op-tical fiber (SMF) to a 2 × 4 fiber optic splitter (Shenzhengigalight, planar lightwave circuit (PLC), wavelength range

Fig. 1. WSI setup showing the tunable laser (TL), 2 × 4 PLC splitter, InGaAs detector, pellicle beamsplitter (PBS), absorber plate (AP), pinhole(PH), aperture stop (AS), sample (S), lenses (L1–L7), optical fibers (OF1, OF2, OF3, OFR), and personal computer (PC).

S. Chakraborty and P. Ruiz Vol. 29, No. 9 / September 2012 / J. Opt. Soc. Am. A 1777

1260–1650 nm, SMF) that divides power equally between thefour output channels into SMF fibers (the second input chan-nel is not used). One output channel, OFR, feeds the referencearm of the interferometer while the other three, OF1, OF2 andOF3, provide three illumination beams. These are collimatedby near-infrared double achromats L1, L2, and L3, and ar-ranged so that they illuminate the sample, S, from three non-coplanar directions. The beams subtend zenith angles θ1, θ2,and θ3 to the optical axis of the imaging system formed bylenses L6 and L7 and are arranged symmetrically around itwith azimuth angles ξ1, ξ2, and ξ3 as shown in the insert inFig. 1. Upon scattering on sample S, reference and objectbeams are combined with a 45∶55 pellicle beam splitterand detected with a near-infrared InGaAs photodetectorarray (SU64SDV-1.7RT/RS170, Goodrich Corporation, 640×512 pixels, 14 bit). Lenses L6 and L7 form a “4f ” double tele-centric imaging system. An aperture stop (AS) serves to con-trol telecentricity, speckle size, and throughput. In thereference beam, a pinhole, PH, is used as a spatial filter andalso to control the intensity ratio between reference and ob-ject beams.

A critical detail in this setup is the way in which the opticalpath lengths are adjusted for the reference and object beams.The reference beam optical path is adjusted with a delay line,while those of the object beams are adjusted by positioningcollimating lenses L1, L2, and L3 at slightly different distancesfrom the sample. If we consider a sphere centered at the in-tersection between the sample surface and the optical axis ofthe imaging system, then the optical center of L1 lies withinthe sphere, of L2 on the sphere surface, and of L3 outsideit. If we assume equal optical paths within the optical fibersOF1, OF2, and OF3 then the distances from the collimatinglenses to the sphere determine their relative optical path dif-ferences, which are used to separate information from differ-ent sensitivity directions in the frequency domain. This isdescribed in detail in the next section. The setup describedabove is effectively a Mach–Zehnder interferometer with mul-tiple illumination directions that are frequency multiplexed byoffsetting their optical path delays.

3. FULL-SENSITIVITY PHASE-CONTRASTWAVELENGTH SCANNINGINTERFEROMETRYA. Working PrincipleConsider a plate made of a semitransparent scattering materi-al that is illuminated and observed using the interferometershown in Fig. 1. The observation direction is defined as thez axis. The plate’s surface is imaged on the photodetector ar-ray with magnification M � 1. The z component of points re-presented by position vectors rs and r measures the distancebetween the reference wavefront and the surface and a pointwithin the material, respectively. A detailed ray diagram isshown in Fig. 2(a) for one illumination beam on the planeof incidence. A point with coordinates r � �x; y; z� in the ma-terial is imaged onto a pixel with indices �m;n� of the photo-detector array. px, py are defined respectively as the pitchbetween pixels along the n and m axes on the photodetectorarray so that x � npx and y � mpy. The optical path differ-ence between light from the pth illumination beam scatteredat point �x; y; z� within the material and the reference wavecan be written as

Λp�x; y; z� � n0�ep · �r0 − rp� � ep · �re − r0�� n1�ep0 · �r − re� � eo · �rs − r�� n0eo · rs; (1)

where n0 and n1 are the refractive indexes of the surroundingmedium (air in this case) and the material, respectively. ep, ep0and eo � �0; 0;−1� are unit vectors along the pth illuminationdirection within the surrounding medium, the illumination di-rection within the material, and the observation direction, re-spectively. r, r0, re, rs, and rp are respectively the positionvectors of points �x; y; z� within the material, imaged ontopixel �m;n�; �x0; y0; z0�, which is imaged onto pixel (0, 0);�xe; ye; ze� where light reaching point r in the material is re-fracted at the surface; �xs; ys; zs� on the material’s surface,which is also imaged onto pixel �m;n�; and finally of�xp; yp; zp�, which locates a point on the illumination wave-front propagating with direction ep that reaches the objectat r0. The triangle with re − r0 as the hypotenuse is a right tri-angle, and all incoming vectors are parallel. The first term onthe right-hand side of Eq. (1) represents the optical path froma point on the illumination wavefront, rp, to point re, wherelight enters the material. The second term represents the op-tical path due to propagation within the material upon refrac-tion, from point re to r and back to the surface at rs. Finally,the last term represents the optical path from the object sur-face at rs to the reference beam’s zero delay wavefrontat z � 0.

The relationship between rs and re in a general case isgoverned by refraction at re by

jrs − rej � jr − rsj tan�θp0 � (2)

and the Snell’s law in vector form:

n0�N × ep� � n1�N × ep0 �; (3)

where θp and θp0 are the incidence and refracted angles of thepth beam, N is a unit vector normal to the surface, (0, 0, 1)here, and ep0 is a unit vector pointing along the refracted beam.

Fig. 2. Generalized optical path diagram for WSI with multiple illu-mination directions for a (a) scattering material and an (b) opaquesurface.


This vector representation is convenient when dealing withnoncoplanar illumination and observation directions, espe-cially at the data processing stage. In the following analysis,we will consider the case of a flat sample that lies perpendi-cular to the observation direction to eliminate extra terms thatwould appear due to refraction at the surface. The generalcase is described in section 2.3.1.4 in [30].

It is convenient to model the object as a set of Ns discretethin scattering layers parallel to its surface. The intensity dueto the interference of the three object wavefronts and the re-ference wavefront can be written as

I�m;n; t� ��A0�m;n; t��

X3p�1

XNs

j�1

Apj�m;n; t�exp�iϕpj�m;n; t��2

.

(4)

A0 represents the amplitude of the reference wavefront. Apj

(p � 1, 2, 3; j � 1; 2;…Ns) is the amplitude of the wavefrontscattered at the jth slice when illuminated by the pth beamand i �

��−1

p. ϕpj is the phase difference between light scat-

tered at the jth slice when illuminated by the pth beam and thereference beam. The spatial indices m and n take the valuesm � 0; 1; 2;…; Nm − 1; n � 0; 1; 2;…; Nn − 1, and t is anondimensional time defined as the true time divided bythe camera interframe time. Expanding Eq. (4) and droppingthe �m;n; t� dependence in all the amplitude and phase vari-ables for clarity gives

I�m;n; t� � A20 �

X3p�1

XNs

j�1

A2pj � 2

X2p�1

X3q�p�1

XNs−1

j�1

XNs

k�j�1

ApjAqk

× cos�ϕpj − ϕqk� � 2A0

X3p�1

XNs

j�1

Apj cos ϕpj: (5)

The right-hand side of Eq. (5) consists of four terms. Thefirst two correspond to a slowly varying function of timedue to both the envelope of the tunable laser power spectrumand to the wavelength dependence of the scattered amplitude.Term 3 represents interference between the light from the jthand the light from the kth layers corresponding to differentillumination beams indicated by indexes p and q, and togetherwith Term 2 are referred to as the autocorrelation terms. Term4 represents the interference between the scattered light fromthe jth slice when illuminated by the pth beam and the refer-ence beam, and is referred to as the cross-correlation term.This latter term contains the information about the micro-structure of the sample.

The phase ϕpj changes with time according to

ϕpj�m;n; t� � ϕsj�m;n� � k�t�Λpj�m;n�; (6)

where k�t� is the wavenumber 2π ∕ λ�t�, and ϕsj is a phase shiftthat may arise at zero nominal path difference due to, forinstance, a phase change on reflection or due to the micro-scopically random arrangements of scatterers contributing tothe amplitude at pixel �m;n� at the jth slice, i.e., the phasethat leads to speckle noise. Λpj represents the optical pathdifference given in Eq. (1) due to the pth beam and for thejth slice, which lies at a depth z � z0 � �j −½�δz below thesample’s surface, δz being the layer thickness. WSI involveschanging k with time over a total range Δk while an image

sequence is recorded. In the ideal case of a linear variation ofk with t,

k�t� � kc � δkt; (7)

where kc � 2π ∕ λc is the central wavenumber correspondingto the central wavelength λc, δk is the interframe wavenum-ber increment, and t ranges from −Δk ∕ 2δk to �Δk ∕ 2δk witha total number of Nt intervals. Substitution of Eq. (7) intoEq. (6) leads to

ϕpj�m;n; t� � ϕsj�m;n� � kcΛpj�m;n� � δkΛpj�m;n�t: (8)

The linear variation of ϕpj with t leads to temporal frequen-cies (units of “cycles per frame”)

fΛpj�m;n� � δkΛpj�m;n� ∕ 2π; (9)

that is, frequencies proportional to the optical path differencebetween light from the pth beam scattered at the jth layer andthe reference wavefront. It is convenient to define an alterna-tive frequency in units of “cycles per scan duration”:

fΛpj�m;n� � fΛpjNt � ΔkΛpj�m;n� ∕ 2π: (10)

Thus, measuring frequencies fΛpj or fΛpj in a pixel-wise ba-sis provides a measure of Λpj�m;n�, which can be used to lo-cate the coordinates of scatterers on the surface and the bulkof the sample. Moreover, if two scans are performed beforeand after a deformation of the sample, then provided ϕsj

remains the same between scans, the phase changesΔϕpj�m;n; 0� provide a direct measure of the change in opti-cal path lengths ΔΛpj�m;n� due to the deformation.

Both the position and phase change of surface scattererscan be evaluated by a time-frequency analysis. The Fouriertransform of the interference intensity signal I�m;n; t� inEq. (5) can be written as an infinite sum over an integer indexthat locates identical spectra Nt samples apart in the fre-quency domain. The goal of the measurement is to reconstructthe amplitude Apj�m;n� and the phase ϕpj�m;n� 3D distribu-tions of the scattered light field from each of the Ns scatteringlayers. If no aliasing is present, then the Fourier transform ofthe intensity in Eq. (4) may be written as

~I� f � � ~W� f � ��

I0 �X3p�1

XNs

j�1

Ipj

�δ� f �

�X3p�1

XNs

j�1

��I0Ipj

qexp��iϕ0pj�δ� f∓fΛpj�

�X2p�1

X3q�p�1

XNs−1

j�1

XNs

k�j�1

��IpjIqk

q

× exp��i�ϕ0pj − ϕ0qk��δ� f∓� fΛpj − fΛqk��; (11)

where � represents convolution, ~W� f � is the Fourier trans-form of W�t�, a continuous window function that representsthe finite sampling duration and the envelope of the laserpower spectrum. The first term between curly brackets inEq. (11) is part of the autocorrelation term and represents


the DC term due to the reference and scattered wavefronts.The second, known as the cross-correlation term, representsthe interference between the reference and all three illumina-tion beams scattered at all layers within the sample and is gi-ven by the superposition of Dirac deltas δ that fall withinbands in the frequency domain between frequencies fΛp1

and fΛpNs which are, according to Eq. (10), proportional tothe optical paths Λp1 and ΛpNs corresponding to the firstand last layers. The phase at the origin of each of these fre-quency components is given by ϕ0pj � ϕpj�m;n; 0�. The thirdterm, known as autocorrelation, represents the interferencebetween pairs of scattering layers corresponding to eitherthe same illumination beam or different ones. It is representedby Dirac deltas at frequencies ��fΛpj − fΛqk�. In order for allthe cross-correlation frequency bands to be fully separated inthe frequency domain and avoid overlap with the autocorrela-tion terms, the carrier frequencies fΛpj and (fΛpj − fΛqk) mustbe carefully set. This is done by adjusting the optical pathsΛpj

by moving the collimating lenses L1, L2, and L3, which willshift the wavefront represented in Eq. (1) by position vectorrp. The second term in Eq. (1) thus adjusts a “piston” term inthe optical path difference Λpj and is characteristic of eachillumination beam.

B. Range and Resolution of the Optical Path DifferenceMeasurementThe maximum unambiguous range that the optical path differ-ence may take is given by the Shannon sampling theorem,which states that in order to ensure adequate sampling of theI�m;n; t� signal, the phase difference ϕpj should not change bymore than π between successive t values, i.e., 2 samples percycle. This leads to a maximum allowed optical path difference

ΛM � π

δk: (12)

The resolution to which the optical path length may be mea-sured can be characterized by the width of the peaks in theFourier domain, given by the width of ~W� f �. A usual resolu-tion criterion is a frequency difference between two neighbor-ing peaks of at least twice the distance from their centers totheir first zero. For the case of normal observation with a tele-centric system as the one illustrated in Fig. 2, the width of thespectral peak and hence the optical path resolution can bewritten as

δΛ � γ2πNtδk

; (13)

in the case of a scattering point in air, and

δΛ � γπ

n1 cos�θp0 ∕ 2�Ntδk; (14)

in the case of scattering points within a medium of refractiveindex n1. We are assuming a plane air/medium interface atwhich refraction occurs, θp0 is the refracted angle and the con-stant γ takes the value 2 for a rectangular window and 4 for aHanning window, for example.

4. IMAGE RECONSTRUCTIONPERFORMANCEA flat scattering surface (sandblasted aluminium plate) wasfirst studied using the interferometer shown in Fig. 1.

The purpose of this was to assess the performance of thelaser and the near-infrared (NIR) camera in terms of thelinearity of the wavenumber scan k�t� and the intensitynoise, which ultimately determine the width of the peak

that represents the surface in the Fourier transform ~I� f �,i.e., the optical path resolution δΛ. This is presented inSubsection 4.A.

Furthermore, it turns out that the scattering surface ap-pears tilted due to the oblique illumination beams that intro-duce a linear spatial variation of the optical path difference inEq. (1), which in the case of a scattering surface reduces to

Λp�x; y; z� � n0�ep · �r0 − rp� � ep · �r − r0� � eo · r�; (15)

with r the position vector of point �x; y; z� now representing apoint on the surface [see Fig. 2(b)]. As the orientation of thesetilted planes in the frequency domain is directly related to theorientation of the illumination beams, they can be conveni-ently used to calculate the illumination unit vectors ep shownin Fig. 1 and therefore the sensitivity vectors that link mea-sured phase changes to actual 3D displacement fields. Thisis described in Subsections 4.B and 4.C.

A. Depth Range and Depth ResolutionA sequence of 8748 frames was recorded during a laser fre-quency scan from ν1 � 237.9305 THz to ν2 � 220.4356 THz(ν � c ∕ λ � ck ∕ 2π, with c the speed of light in vacuum). Eachframe was 64 × 64 pixels, encoding the intensity I�m;n; t� in14 bits. The field of view was ultimately limited by the randomaccess memory available for data processing, but 512 ×640 pixels (rows, columns) images can be acquired withthe NIR camera. Fourier transformation of the recorded 3Dintensity distribution is performed on a pixel-wise basis alongthe t axis, leading to a spectrum that is directly related to theposition of the surface along the observation direction at co-ordinates x � npx, y � mpy. The tunable laser delivers anearly constant power during the whole scan, which trans-lates in a good level of signal modulation. Figure 3(a) showsthe intensity for a pixel at the center, plotted against framenumber, i.e., I�32; 32; t�. The mean value of the intensity signalwas subtracted to eliminate the dominant DC peak at fΛp � 0.Also, the intensity signal was multiplied by a Hanning windowW�t� � 0.5f1 − cos�2π�t� Nt ∕ 2� ∕Nt�g, −Nt ∕ 2 ≤ t ≤ �Nt ∕ 2 toreduce spectral leakage between the otherwise prominentsecondary lobes of a rectangular window ~W� f �. The Fouriertransform of the signal in Fig. 3(a) is shown in Fig. 3(b) in thepositive frequency axis and reveals six peaks. The frequencyaxis was converted to optical path difference by usingEq. (10). Peaks are labeled according to which pair of beamsinterfere; for instance, “01” indicates interference between thereference beam and object beam 1 and “23” indicates interfer-ence between object beams 2 and 3 (see insert in Fig. 1). No-tice in Fig. 3(b) how peaks “01,” “02,” “03” are fully separated.This separation is adjusted by moving lenses L1, L2, and L3toward or away from the sample so that the peaks do notoverlap with the cross interference ones and no aliasing isintroduced. Figure 3(c) shows peak “01” in more detail. Ithas a width of ∼0.06 mm, which compares well with theexpected value of 0.068 mm from Eq. (13).

When this computation is done for every pixel and the crossinterference between the object beams are neglected, an in-stance of the surface emerges for each object beam within


the so-called “reconstruction volume” in the “direct,” i.e.,�x; y; z�, space. As mentioned before, the surface appears withdifferent tilts for each illumination direction. Figure 4 shows aslice through the reconstructed volume parallel to the yz planefor x � 32. The three lines to the left correspond to the crossinterference between the object beams. The three on the rightcorrespond to the interference between the object beams andthe reference beam, and thus represent the object as recon-structed for each illumination direction.

B. Evaluation of Illumination and Sensitivity VectorsThe tilt of the reconstructed surface is uniquely related to theillumination directions. This allows us to use them to calculatethe orientation of the illumination beams ep. Figure 5(a)shows the planes of best fit for all surface reconstructions“01,” “02,” and “03.” The normal to reconstruction “01,” calcu-lated from the coefficients of the plane of best fit, is shown inFig. 5(b). It is convenient to denote the normal vector to thepth reconstructed surface by np, and express it in terms of αpand βp, the zenith and azimuth angles that np subtends fromthe observation axis and the x axis, respectively. It can beshown that

np � �sin αp cos βp; sin αp sin βp; cos αp�; (16)

or equivalently,

αp � cos−1� k · np�; (17)

βp � tan−1�j · npi · np

�; (18)

where i, j, and k are unit vectors along the x, y, and z axes,respectively.

From Fig. 2(b) it follows that the optical path differencebetween points r0 and r is Λ�r� −Λ�r0� � n0ep · �r − r0� �n0jr − r0j sin θp, with θp the zenith angle of ep to the observa-tion direction. If n0 � 1 then the optical path differencerepresents geometrical distance, i.e., Δz � jr − r0j sin θp. Asthe surface normal vector np subtends an angle αp to the zaxis [Eq. (16)], then

tan αp � jr − r0j sin θpjr − r0j

� sin θp: (19)

Similarly, it can be shown that the azimuth of the normalvector, βp, is simply related to the azimuth of correspondingillumination vector, ξp, by

ξp � βp � π: (20)

Therefore, once the normal vectors np have been obtainedfor all surface reconstructions via plane fitting, the angles αpand βp are obtained from Eqs. (17) and (18) and finally theillumination direction ep is determined using Eqs. (19) and(20) as

ep � �sin θp cos ξp; sin θp sin ξp; cos θp�: (21)

The sensitivity vectors of the system can now be evaluateddirectly using Eq. (21) and the observation vector eo ��0; 0;−1� (see Fig. 2):

Fig. 3. (a) Intensity signal recorded at 1 pixel during a WSI scan,when an opaque flat surface is imaged under three-beam illumination;(b) its corresponding Fourier transform, showing cross correlationterms “01,” “02,” and “03” and autocorrelation terms “12,” “13,” and“23”; (c) peak 01 in more detail.

Fig. 4. yΛ cross section of the magnitude of the Fourier transformvolume obtained when a flat opaque surface is reconstructed, shownin reverse contrast for clarity. Lines corresponding to the auto corre-lation terms “12,” “23,” and “13” and the cross-correlation terms “01,”“02,” and “03” are clearly visible. Their tilt is a consequence of theoblique illumination.

Fig. 5. (a) Planes of best fit obtained for reconstructions “01,” “02,”and “03” for a reference flat surface; (b) plane of best fit for recon-struction “01” showing its normal vector; and (c) sensitivity andobservation vectors, as evaluated from the reference surface recon-structions using Eqs. (16)–(22).


Sp�2πλc

�ep−eo��2πλc

�sin θp cos ξp;sin θp sin ξp;1�cos θp�;

(22)

or, in matrix form,

S � 2πλc

0@ sin θ1 cos ξ1 sin θ1 sin ξ1 1� cos θ1sin θ2 cos ξ2 sin θ2 sin ξ2 1� cos θ2sin θ3 cos ξ3 sin θ3 sin ξ3 1� cos θ3

1A: (23)

The sensitivity and observation vectors, as evaluated fromthe reference surface reconstructions using Eqs. (16)–(22) areshown in the Fig. 5(c).

C. 3D Displacement DistributionThe technique presented in this paper has been designed tomeasure all displacement components in the full volume ofscattering materials with interferometric sensitivity. This isdone by measuring changes in the 3D phase distributions be-tween two separate scans before and after loading or movingthe sample. By using three illumination directions it is possibleto simultaneously evaluate the phase change at every pointwithin the reconstructed volume. The relationship betweenthe measured phase change and the displacement field canbe expressed in matrix form by

Φ � SΔr; (24)

where Δr � �u; v;w�T and Φ � �Δϕ1;Δϕ2;Δϕ3�T (T indi-cates transposed) and where u, v, and w represent the displa-cements along the x, y, and z axes, respectively. Finally, the3D distribution of the displacement vector within the sampleis thus obtained through inversion of the sensitivity matrix as

Δr � S−1Φ: (25)

This operation has to be performed in a voxel-by-voxelbasis in the reconstructed volume.

D. Volume RegistrationIn order to solve Eq. (25) for each voxel in the reconstructedvolume, the reconstructions associated to different illumina-tion beams, i.e., peaks “01,” “02,” and “03,” need to be “regis-tered.” Registration is the process by which correspondingvoxels in each reconstructed volume are brought to the sameposition in a common coordinate system. This usually in-volves a mathematical transformation that may include a com-bination of translation, rotation, and shear in most cases. Inthis paper, registration requires the subtraction of the carrierfrequency for each illumination direction described by thefirst term between squares brackets in Eq. (15). Moreover,subtraction of the position-dependent optical path differencedescribed in the second term between square brackets inEq. (15), ep · �r − r0�, is also required. Fortunately, this is a sim-ple task once the normal vectors np have been estimated, andcan be done efficiently through a simple geometric transfor-mation. Figure 6 shows a diagram of the data processing stepsfor both the reference surface and the volume sample.

5. 3D FULL-SENSITIVITY DISPLACEMENTMEASUREMENTSIn this Section, we present experimental results on the mea-surement of a 3D displacement field within the bulk of anepoxy resin block seeded with scattering particles (titaniumoxide, 1 μm average diameter) much smaller than the sizeof the volume point spread function (PSF) of the WSI system(25 μm laterally and 68 μm axially). The particles/resin volumefraction was ∼3 × 10−3, which results in ∼100 particles per

Fig. 6. Schematic diagram of the data processing for the reference surface and a volume sample.


PSF. If particles were sparse as required by particle image ve-locimetry, there would be regions in the volume that wouldnot contribute any signal and thus no displacement informa-tion could be retrieved. On the other hand, a high volume frac-tion would result in multiple scattering, which increases thenoise floor and reduces the depth resolution of the system(the point spread function broadens in the axial direction).This is because of the extended optical path due to multiplescattering events, which effectively delocalizes the scatteringcenters.

The displacement field consists of in-plane rotation on thex-y plane (see Fig. 1) and out-of-plane tilt about the x axis,which were introduced with a rotation and a kinematic stage,respectively. These controlled displacements are easy to im-plement and provide a benchmark to validate the full-fieldmeasurements obtained with the WSI system. Figure 7 shows“yΛ” cross sections of the registered reconstructed volumesof the sample, “01,” “02,” and “03,” corresponding to each ofthe illumination beams. The top row shows the magnitude ofthe Fourier transform, while the bottom row shows thewrapped phase difference volumes Δϕ1, Δϕ2 and Δϕ3 ob-tained after rotating and tilting the sample (phase values be-tween −π and π). A cubic kernel of 7 × 7 × 7 voxels was usedon the terms sin�Φ� and cos�Φ� before the atan functionevaluation, to reduce speckle noise. The phase volumes werethen unwrapped with a 3D path following algorithm that pre-vents the unwrapping path to go through phase singularityloops [31,32]. Finally, the 3D displacement field componentsu, v, and w were calculated for each voxel using Eq. (25). Thepresence of some unwrapping errors in theΔϕ1,Δϕ2 andΔϕ3

phase volumes meant that only 16 × 35 × 71 �m;n; p� voxelswere obtained without unwrapping errors, corresponding toa volumetric field of view of ∼0.9 mm × 0.4 mm × 0.9 mm�x; y; z�. Figure 8 shows cross sections of the measured aver-age 3D displacement field corresponding to the epoxy sampleunder in-plane rotation and out-of-plane tilt. From top to bot-tom, the rows indicate the displacement components u, v, andw along the x, y, and z axes, respectively. From left to right,the columns show sections of the data volume on planes yΛ,xΛ, and xy. Notice the gradient in the u and v displacement

fields, as expected for an in-plane rotation around the obser-vation direction, and the gradient of displacement w at thebottom right, which corresponds to out-of-plane tilt. In orderto compare the measured displacements with the known onesintroduced with the rotation and tilt stages, the u�x; y�, v�x; y�,and w�x; y� displacements shown in Fig. 8 were averagedalong the x, y, and x axes, respectively, to reduce noise. Inall cases an excellent agreement was observed between the“measured” and “reference” displacements (see Fig. 9). Theroot mean squared deviation between the theoretical andmea-sured displacements u�x; y; z�, v�x; y; z�, and w�x; y; z� were,respectively, 0.14 μm, 0.20 μm, and 32.5 nm.

Assuming no memory constraints during image acquisition,the current WSI interferometer can generate a raw datavolume of approximately 512 × 640 × Nt (rows, columns,frames). The maximum number of frames Nt is given bythe ratio of the laser frequency tuning range to the minimum

Fig. 7. Cross sections of magnitude and the wrapped phase volumesof a scattering sample that has undergone simultaneous in-plane rota-tion and out-of-plane tilt (phase values between −π and π).

Fig. 8. (Color online) Cross sections of the measured 3D displace-ment field corresponding to a sample under in-plane rotation andout-of-plane tilt. The rows indicate the displacement components u,v, and w along the x, y, and z axes, respectively. The columns showsections of the data volume on planes yΛ, xΛ, and xy. Displacementsunits in meters.

Fig. 9. “Measured” (bold line) and “reference” (dotted line) averagedisplacement profiles obtained for simultaneous in-plane rotation andout-of-plane tilt of the sample.


frequency step: Nt � Δν ∕ δν � 17.495 THz ∕ 0.0002 THz∼87; 000 frames. After re-registration, the size of the “displace-ment” volumes u, v, and w is reduced to ∼512 × 640 × Nt ∕ 24due to the data processing required in the Fourier domain. Thefactor 1 ∕ 24 arises from the fact that ½ of the frequency axis(negative frequencies) is neglected, another ½ is lost to theautocorrelation terms, then ⅓ of the remaining bandwidthis used for each sensitivity vector and finally ∼½ of that rangeis effectively used to accommodate the object thickness dueto the tilt in the frequency space. Thus, the reconstructed dis-placement volumes would contain ∼512 × 640 × 3600 voxels.If a 7 × 7 × 7 convolution kernel is used to reduce spatial noisethen the displacement volumes would provide ∼70 × 90 × 520independent measurements. The maximum “effective” depthrange can be estimated from Eq. (12) by considering δk �4.19 m−1 and an extra factor of 1 ∕ 12 (½ already taken intoaccount by discarding negative frequencies) as ΛM eff �ΛM ∕ 12∼ 62 mm (6.2 mm in the current setup as only 8748frames were recorded, as explained in Subsection 4.A). Thisis assuming that multiple scattering and absorption do not seta lower limit.

6. CONCLUSIONWe described a method based on wavelength scanning inter-ferometry to measure 3D displacement fields within scatteringmaterials. It provides 3D reconstructions of the material mi-crostructure and also depth-resolved phase information thatis used to evaluate all the components of the displacementvector at each voxel of the data volume, effectively extendingfull-sensitivity digital speckle pattern interferometry intothree dimensions. The ability to measure the complete setof displacement components within the volume of the sampleopens exciting opportunities in experimental mechanics; forinstance, the identification of constitutive parameters in ani-sotropic or multimaterial samples. This is usually done by fi-nite element model updating, or alternatively using the virtualfields method, two powerful inversion techniques that requirefull field displacement or strain fields. Some issues that needto be investigated include the effects that material dispersion,birefringence, optical absorption, multiple scattering, and sur-face refraction may have in the measured displacements andstrains. Moreover, phase singularities can be so abundant andcomplex in a 3D volume that robust 3D phase unwrapping al-gorithms are required to expand the measurement volume.This technique can be viewed as frequency multiplexedOCT in which each channel carries information for a specificdisplacement sensitivity.

ACKNOWLEDGMENTSThe authors wish to thank the Engineering and PhysicalSciences Research Council (EPSRC) for their financial sup-port with grant EP/E050565/1, Dr. G. E. Galizzi for his techni-cal guidance, and Prof. J. M. Huntley for his assistance withthe 3D unwrapping software.

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