Tilt Scanning Interferometry

8/2/2019 Tilt Scanning Interferometry

1/23

doi: 10.1098/rspa.2006.1685, 2481-25024622006Proc. R. Soc. A

Pablo D Ruiz, Jonathan M Huntley and Alejandro Maranonmediadisplacement fields within optically scatteringmapping structure and three-dimensionalTilt scanning interferometry: a novel technique for

References

html#ref-list-1http://rspa.royalsocietypublishing.org/content/462/2072/2481.full.

This article cites 19 articles

Email alerting serviceherethe box at the top right-hand corner of the article or click

Receive free email alerts when new articles cite this article - sign up in

http://rspa.royalsocietypublishing.org/subscriptionsgo to:Proc. R. Soc. ATo subscribe to

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

2/23

Tilt scanning interferometry: a noveltechnique for mapping structure and

three-dimensional displacement fields withinoptically scattering media

BY PABLO D. RUI Z, JONATHAN M. HUNTLEY* AN D ALEJANDRO MARANON

Wolfson School of Mechanical and Manufacturing Engineering, LoughboroughUniversity, Loughborough, Leicestershire LE11 3TU, UK

We describe a novel technique that we call tilt scanning interferometry to measuredepth-resolved structure and displacement fields within semi-transparent scatteringmaterials. The method differs significantly from conventional optical coherencetomography, in that only one wavelength is used throughout the whole measurementprocess. Temporal sequences of speckle interferograms are recorded while theillumination angle is tilted at constant rate. Fourier transformation of the resultingthree-dimensional intensity distribution along the time axis reconstructs the scatteringpotential within the medium. Repeating the measurements with the object wave atequal and opposite angles about the observation direction results in two three-dimensional phase-change volumes, the sum of which gives the out-of-plane-sensitive

phase volume and the difference between which gives the in-plane phase volume. Fromthese phase-change volumes the in-plane and out-of-plane depth-resolved displacementfields are obtained. The theoretical framework for the technique is explained in detailand a practical optical implementation is described. Finally, results from proof-of-principle experiments involving a semi-transparent beam undergoing bending arepresented.

Keywords: speckle interferometry; depth-resolved displacements; whole-field;

displacement measurement; optical coherence tomography;

tilt scanning interferometry

1. Introduction

The ability to measure internal displacements and strains within a material hasmany potential applications in engineering and medical sciences. A broad rangeof methods has been developed in the last few decades, such as neutrondiffraction (ND; Fitzpatrick & Lodini 2003; Hutchings et al. 2005), photoelastictomography (Abe et al. 1986; Aben et al. 2004), phase contrast magneticresonance imaging (PCMRI; Steele et al. 2000; Draney et al. 2002) and three-dimensional digital image correlation (DIC) using data acquired with X-ray

computed tomography (Bay et al. 1999) and optical coherence tomography

Proc. R. Soc. A (2006) 462, 24812502

doi:10.1098/rspa.2006.1685

Published online 30 March 2006

* Author for correspondence ([email protected]).

Received 26 October 2005Accepted 27 January 2006 2481 q 2006 The Royal Society

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

3/23

(OCT; Schmitt 1998; Fercher et al. 2003). Each technique has a restricted rangeof materials to which it can be applied: ND, for example, is suitable forpolycrystalline metals; PCMRI requires significant water or fat content in thesample. For many technologically and medically important materials (opticallyscattering polymers, composites and biological tissues), the existing techniquesare often either not applicable or else are too insensitive. The lack of sensitivity

arises from the fact that in DIC the sensitivity is coupled to the spatial resolutionof the technique and is therefore limited by it.

Recent promising developments in OCT technology sidestep this problem bymaking use of the phase information within the OCT signal, thereby decouplingthe displacement sensitivity and depth resolution (Gastinger et al. 2003; Gulker &Kraft 2003; Ruiz et al. 2004, 2005). As a result, displacement sensitivity isimproved by 23 orders of magnitude compared to the depth resolution of state-of-the-art OCT systems.

The resolution of depth in OCT is achieved through the use of multiplewavelengths which are present either simultaneously (Born & Wolf 1959; Dresel

et al. 1992), leading to a narrow-gated region of modulating scatterers, or elsesequentially. The latter case is known as wavelength scanning interferometry(WSI; Takeda et al. 1982; Fercher et al. 1995; Lexer et al. 1997; Kuwamura &Yamaguchi 1997; de Groot 2000) and relies on a depth-encoding frequency shiftin the interference signal as the wavelength of the light source is swept through anarrow range (see figure 1a). While the WSI form of OCT offers superior signal-to-noise ratio to standard OCT for displacement sensing, it suffers from asignificant practical drawback that tunable light sources are expensive anddifficult to operate without mode-hopping.

In this paper we present a different approach to measuring depth-resolved

displacements within optically scattering materials, which is based on tilting theilluminating beam during the acquisition of the image sequences. This providesthe necessary depth-dependent phase shifts that allow the reconstruction of the

l(t)

l

R

S1

S2

f2 f1

Sj

(a)

R

S1

S2

f2 f1

Sj

(b)

Figure 1. (a) Generation of depth-encoding frequency shifts by: (a) tuning the wavelength of thelight source through a narrow range (wavelength scanning interferometry) and (b) by tilting amonochromatic wavefront (tilt scanning interferometry).

P. D. Ruiz and others2482

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

4/23

object structure and its internal displacements. The method is quite distinct fromall the previous OCT literature, in that only a single wavelength is presentthroughout the entire data recording process, thereby considerably reducing theexpense and complexity of the light source. The depth-encoding frequency shiftcan now be regarded as coming from the Doppler shift of the photons reflected

from the tilting mirror in the object illumination beam path. This is shownschematically and in a simplified way in figure 1b, where different heterodynefrequencies are produced from scattering points lying at different depths withinthe material. As with the WSI version of OCT, displacement sensitivity of thisnew technique that we denote here by tilt scanning interferometry (TSI) isdecoupled from the intrinsic depth resolution of the technique and is a few tens ofnanometres at optical wavelengths. Extraction of depth and displacementinformation of non-transparent object surfaces has been previously reportedusing speckle contouring techniques based on source displacement (Rodrguez-Vera et al . 1992). TSI extends this capability to study depth-resolved

displacement fields inside semi-transparent materials.In 2, we provide a mathematical description of the technique following a ray-tracing approach to model the problem. We also explain how to obtain thescattering potential of the sample as well as depth-resolved displacement fieldsand introduce some important parameters of the technique, namely the depthresolution, gauge volume, depth range and displacement sensitivity. In 3, wepresent a proof-of-principle experiment based on a TSI approach and showencouraging results of depth-resolved displacements measured within a beamunder three-point bending up to a depth of over 5 mm. We compare themeasurements with finite-element predictions, discuss some error sources, andfinally in 4 draw some conclusions.

2. Tilt scanning interferometry

(a) Depth-dependent phase shift introduced by a tilting wavefront

Figure 2a shows a semi-transparent scattering material of refractive index n1immersed in a medium of refractive index n0, illuminated by a collimatedbeam of wavelength l at an angle q to the optical axis of the system. Weassume that the surface is flat, though this restriction can be relaxed through asuitable extension of the analysis presented here. It is convenient to place a

flat but microscopically rough opaque surface over the region y%0 of the planezZ0, where the coordinate system (x, y, z) is as defined in figure 2a. Thisreference surface serves two purposes: first, allowing for correction of thenonlinearity of the tilting device and second (since it does not strain duringthe loading of the sample), enabling registration of the before- and after-loadscans of the sample. We should emphasize, however, that it does not providethe reference wave for the interferometer, which rather is introduced by meansof a separate beam splitter, BS. While the reference surface has the obviousdrawback of obscuring the sample for some of the pixels in the field of view(FOV), it could, in principle, be dispensed with given a sufficiently well-

calibrated tilting device.The illumination beam is refracted at the object surface z1(x, y) and reachespoint F with coordinates (x, y, z) at an angle qrZsin

K1n0sin q=n1 to the optical

2483Tilt scanning interferometry

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

5/23

axis. Some of the light scattered at F travels vertically downwards, and isrecombined by BS with a reference wave derived from the same laser light sourceas the object beam, and is imaged onto a pixel lying within a two-dimensional

photodetector array.The phase difference between light scattered at F and a reference wavefrontwith phase fr can be expressed relative to the phase difference at point G, whichlies on a rough opaque reference surface R at the origin (0, 0, 0), as

fx; y; zZf0; 0; 0C2p

ln0x sin qCn0z11Ccos qCn1zKz11Ccos qr;

2:1

where we assumed that zRz1R0. The phase differences due to the first, secondand third terms between square brackets in equation (2.1) account for the optical

paths AB, CDCDB and EFCFD, respectively. The random distribution ofscattering centres within the material gives rise to a speckle phase distributionalong x in this two-dimensional representation.

If the illumination angle q changes linearly with time about the centre angleqc, i.e.

qtZ qcCDq

Tt; 2:2

T being the time it takes to scan through the tilt angle Dq and t the timevariable (KT/2%t%T/2), then the phase f(x, y, z) will vary as

vfx; y; z; t

vtZ

vf0; 0; 0; t

vtC

2pn0l

Dq

Tx cos qKz1 sin qKzKz1x: 2:3

z1

z

xy

zmax F(x,y,z)

Efz(y,zz1)

f(x,y,z1)fz1(y,z1)

f(x,y,0)

fx(x,y)

f(0,y,0)

(0,y,0)(x,y,0) x

magnitude

I

r(x,y

)

f(x,y,z)

frominside

theobject

fromrefere

ncesurfac

eR(x,y,z1)

(x,y,0)(0,0,0)

R opticalaxis

I(x,y, t)

(a) (b) (c)

n1

n0

qr

fr

q

q

l

DC B RG

A

BS

z

fory>0

fory

8/2/2019 Tilt Scanning Interferometry

6/23

In the last term between the square brackets the following relationship, derivedfrom Snells law of refraction, was used:

vqr

vqZ

c cos q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1Kc2 sin2qp

; 2:4

with cZn0=n1. For convenience, we define a parameter x as

xZc cos q sin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1Kc2 sin2qp : 2:5

Equation (2.3) can be expressed in terms of temporal frequencies as

fx; y; zZf0; 0; 0Cn0Dq

lTx cos qKz1 sin qKzKz1x; 2:6

Zf0; 0; 0Cfxx; yCfz1y; z1Cfzy; zKz1: 2:7

f(0, 0, 0) is a carrier frequency due to the rigid body translation or pistonterm of wavefront GA as it tilts around an axis perpendicular to the plane offigure 2a, and is zero if that axis passes through point G. The frequency ofthe second term in equation (2.7) varies linearly with x, whereas fz1(y, z1)accounts for the distance from the object surface to the reference surface,z1, and fz(y, zKz1) to the depth zKz1 of scattering points within the sample.The last two terms can be interpreted as depth-encoding heterodyne frequencyshifts due to the Doppler shift of the tilting beam. f(x, y, z) in equations (2.6)and (2.7) represents the absolute modulation frequency produced byinterference between light coming from point F at position (x, y, z) within theobject and the plane reference wavefront with phase fr coming from therecombining beam splitter.

At zZz1, fzy; zKz1Z0 and fx; y; z1Zf0; 0; 0Cfxx; yCfz1y; z1 is thefrequency associated with point D at position (x, y, z1) on the object surface infigure 2a. If we put zZz1Z0, fzZy; zKz1Zfz1y; z1Z0 and fx; y; 0Z

f0; 0; 0Cfxx; y is the frequency associated with point B lying at (x, 0, 0) onthe reference surface R in figure 2a.

(b ) Extraction of the scattering potential

The scattering potential of the sample can be extracted from the Doppler

shifts by effectively mapping depth from frequency. In what follows, we willassume, as in standard OCT, that the contributions from multiple scatteringwithin the material can be neglected. The interference between light coming fromall the scattering points along DF in figure 2a and the reference wavefront frgives rise to an intensity signal modulated with multiple frequencies:

Ix;y; tZ Irx;yC2

zmaxz1

Irx; yIx;y; z1=2cos2pfx; y; ztdz

C2

zmaxz1

zmaxz1

Ix;y; zIx; y; z01=2cosf2pfx; y; zKfx; y; z0tgdzdz0:

2:8While the first term in the right-hand side of equation (2.8) represents the dccomponent of the reference beam, the second term corresponds to the

2485Tilt scanning interferometry

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

7/23

modulation due to interference between the reference beam and light scatteredwithin the material. The integration limit zmax represents either the objectback surface, the maximum penetration depth which depends on theabsorbance of the material and the laser power, or the depth range of thesystem (as discussed in 2d), whichever is the minimum. The double integral

in the third term is due to cross-interference between light coming from withinthe object and contributes to the dc component and low-frequency componentsin the interference signal.

The frequency of each term on the right-hand side of equations (2.6) and (2.7)is, in general, dependent on q and therefore changes during the course of the scan.We assume for now that Dq is small enough so that the resulting frequency shiftscan be neglected.

A one-dimensional Fourier transform of signal I(x, y, t) (the light intensitymeasured at a single pixel) along the time axis gives rise to a spectrum as shownschematically at figure 2b. Any given pixel either sees the reference surface or the

sample, but not both; this figure can therefore be interpreted as a top view of thespectrum over all y onto the (f, x) plane, i.e. as a superposition of the spectracorresponding to the reference surface R and the interior of the sample. There isa dc peak at fZ0 for all x, another peak at f(x, 0) corresponding to the referencesurface R, and a band associated with the object. The leading edge at f(x, z1)generally shows the highest amplitude of the band due to the refractive indexchange at the surface of the sample. For higher frequencies, the amplitudedecreases due to absorption and multiple scattering within the material. Theposition of the peaks is therefore associated with the internal structure of theobject and its position relative to the reference surface, whereas their amplitudesare related to the degree of scattering or reflection coefficient at each point withinthe object or at the reference surface.

Usually, the spectrum ~Ix;f (where w indicates a Fourier transformedvariable) is obtained through the Fourier transform of the product of theintensity signal I(x, t) with a window function W(t). In this way, the Fouriertransform of I(x, t) is convolved in the frequency domain with the Fouriertransform of the window function, ~Wf. A Hanning window WtZ1=2K1=2cos2ptCT=2=T was used in this work in order to reduce the cross-talkbetween secondary peaks that would be present if a rectangular window, orequivalently no window at all, were used.

The position of the object surface relative to the reference surface R, z1, is

proportional to fz1, which is the frequency difference between the spectral peakdue to the scattering at the sample surface and that due to the reference surface.This difference is not present in a spectrum from a single pixel, but rather iscalculated from two or more pixels imaging the reference and sample at the samex-value (i.e. from the same column of the photodetector array), so that all otherterms in the right-hand side of equation (2.7) are common to both. From thethird term of equation (2.7), we have

z1ZKlTfz1

n0Dq sin q: 2:9

For simplicity, the following analysis will be limited to the case where z1(x)is a constant. For objects with a surface of arbitrary shape, the refracted angleqr at point F would depend on the coordinates x, y, z and the angle between

P. D. Ruiz and others2486

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

8/23

the incident illumination beam and the normal of the object surface to theoptical axis at the point where the ray that ends in F intersects the surface. Inthis case, the expression for the optical phase difference f(x, y, z) in equation(2.1) would include extra terms. This case can be simplified in two differentways:

(i) Experimentally, by immersing the object in a cell filled with indexmatching fluid. This approach is equivalent to having a flat objectsurface which is parallel to the reference surface R.

(ii) Numerically, by solving the refraction problem in two stages. Theproblem of an object with a flat surface parallel to the reference surfaceis solved first and then the refraction effects due to a curved surface (orarbitrary surface) are evaluated to correct the first approximation.

For the initially flat samples used in the experiments presented here, suchcorrections were relatively minor and neglected for the rest of the paper.

Once z1 has been evaluated through equation (2.9), the position z of ascattering point underneath the object surface can be obtained as

zx; yZ z1KlTfx; y; zKfx; y; z1

xn0Dq: 2:10

The difference between the terms in square brackets is simply fzy; zKz1(as seen in figure 2b) and depends on the refractive indices of the material andsurrounding medium and on the illumination angle through the parameter x.

The spectral bandwidth Df associated with a thickness Dz within the objectcan be obtained using equation (2.10) as

DfZKn0jxjDq

lTDz: 2:11

z < z1

8/2/2019 Tilt Scanning Interferometry

9/23

This is illustrated in figure 3, with an object of constant thickness and differentrefractive index for different values of x. The spectrum looks narrower for thematerial with higher refractive index n2, for which x is lower.

The scattering potential is obtained simply by a mapping of the modulationamplitude j~Ix; y;fj2 from spectrum coordinates (x, y, f) into spatial coordinates(x, y, z) through the relationship between frequency and position shown inequation (2.10).

While a single illumination direction is sufficient to obtain the scatteringpotential, multiple directions can be used to improve its estimation.A symmetric lateral illumination configuration has the advantage of providingspectra with the same bandwidth for a given object thickness, assuming auniform refractive index distribution. If the left and right illumination anglesqcZqL and qcZqR are set so that jxj is a maximum, then small differencesbetween qL and qR will produce negligible differences between jxqLj andjxqRj, thus relaxing alignment requirements. A more important consequenceof this choice of qL and qR is that the depth resolution of the system is alsooptimized. Figure 4 shows the variation of jxqj with q for a range of typical cvalues. The variation with refractive index means that the depth resolution

outside the object is better than that within it.

(c) Depth-resolved displacements

The position of scattering points within the object will change according to themechanical properties of the material and the loads applied to it. The value offat a particular voxel (x, y, z) in the specimen will reflect that change and can becalculated from the real and imaginary parts of the spectrum ~Ix; y;f at afrequency given by equation (2.10).

For the system shown in figure 5, displacements in the y direction cause no

phase change due to the illumination geometry. If point F at (x, y, z) moves toa new position F0 with coordinates (xCu, y, zCw) after deformation (seefigure 6), then from equation (2.1) the phase difference after displacement can

20 40 60 80

incidence angle q(deg)

0

0.2

0.4

0.6

0.8

1.0

x

1.0

1.4

1.21.3

1.1

Figure 4. Variation of refraction parameter x with the illumination angle q for five different values

ofcZn0=n1Z1; 1:1; 1:2; 1:3 and 1:4.

P. D. Ruiz and others2488

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

10/23

be written as

fxCu;y;zCwZf0;0;0C2p

lfxCux;y;zn0 sinqCz1Cwx;y;z1n0

!1CcosqCzCwx;y;zKz1Kwx;y;z1n11Ccosqrg:

2:12

While a single illumination direction is sufficient to extract the scatteringpotential, at least two illumination directions are essential to determine the in-plane (x) and the out-of-plane (z) displacement components. For right and leftlateral illumination, where qRO0 and qL!0, respectively, the phase differenceDfx; y; zZfxCu; y; zCwKfx; y; z due to object deformation is

DfRx; y; zZ2p

lfux; y; zn0 sin qRCwx; y; z1n01Ccos qR

C wx; y; zKwx; y; z1n11Ccos qRrg;

DfLx; y; zZ2p

lfKux; y; zn0 sin qLCwx; y; z1n01Ccos qL

C wx; y; zKwx; y; z1n11Ccos qLrg;

9>>>>>>>>>=>>>>>>>>>;

2:13

where w(x, y, z1) is the out-of-plane displacement of point D at (x, y, z1) to D0

at (x, y, z1Cw(z1)). By choosing qRZKqLZq, the in-plane (x) and out-of-plane(z) phase difference components Dfxx; y; zZDfRx; y; zKDfLx; y; z and Dfzx; y; zZDfRx; y; zCDfLx; y; z are obtained as

Dfxx; y; zZ4p

ln0ux; y; zsin q; 2:14

L1

L2

L3

WBSC

O

M2

M1

CBS

L4TM

S

OF2

OF1

R

RG

q q

Figure 5. Optical setup of a tilt scanning interferometer showing: optical fibres OF1 and OF2; lensesL1L4; tilting mirror TM; cube beam splitter CBS; mirrors M1, M2; alignment screen S; wedgebeam splitter WBS; camera C; ramp generator RG and object O.

2489Tilt scanning interferometry

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

11/23

Dfzx; y; zZ4p

lfwx; y; z1n01Ccos qKn11Ccos qr

Cwx; y; zn11Ccos qrg: 2:15

The first term within the {.} in equation (2.15) corresponds to the phase changedue to a displacement of the object surface relative to the reference surface, whilethe second one is due to the displacement of point F to F 0 along the optical axis.The in-plane and out-of-plane displacements can be extracted from equations(2.14) and (2.15) as

ux; y; zZlDfxx; y; z

4pn0 sin q; 2:16

wx; y; zZlDfzx; y; z

4pn11Ccos qrKwx; y; z1

n01Ccos q

n11Ccos qrK1

: 2:17

Putting zZz1 into equation (2.17) gives

wx; y; z1ZlDfzx; y; z1

4pn01Ccos q: 2:18

Substitution of equation (2.18) into equation (2.17) then leads to

wx; y; zZlDfzx; y; z

4pn11Ccos qr

KlDfzx; y; z1

4p

1

n11Ccos qrK

1

n01Ccos q

: 2:19

This suggests that in the case of a sample with a lay-up of different refractiveindexes, or index gradient in the depth dimension, the displacements evaluatedfor shallow slices should be used to correct for the displacements at deeper ones.

In the case of rigid body translation of the sample along Cz, then wx; y; zZwx; y; z1 and equation (2.15) reduces to the standard expression for out-of-planesensitivity (Huntley 2001).

x

yz

dx

dz

d

w(x,y)

u(x,y)

d

F

F'

Figure 6. A slice through the gauge volume in the tilt scanning interferometry system showingdepth resolution dz, lateral resolution dx and average displacement d of scatterers.

P. D. Ruiz and others2490

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

12/23

(d) Gauge volume, depth range and displacement sensitivityIn a symmetric lateral illumination system setup, the measurement volume is

defined by the intersection of the illumination beams during the whole tilt scan(see figure 7). The depth resolution can be defined as the minimum distancebetween surfaces inside the measuring volume whose corresponding interferencesignals can be fully resolved in the frequency domain. The usual resolutioncriterion is that the frequency difference df between two neighbouring peaks hasto be at least twice the distance from their centres to their first zero, which isequivalent to the width of the central lobe. If a window function W(t) is used,apart from reducing the leakage through secondary lobes, it has the effect of

broadening the spectral lines. A rectangular window of duration T, for example,results in a sinc function of width dfZ2/T, while a Hanning window has aspectral width ofdfZ4/T. From equation (2.11), the depth resolution is therefore

dzZKgl

n0jxjDq; 2:20

where gZ2, 4 for rectangular or Hanning windows, respectively. Although therectangular window has superior depth resolution, this is accompanied by theundesirable presence of large secondary lobes in ~Wf, which may interfere with

other peaks, leading to phase errors. This effect is significantly reduced if aHanning window is used instead.

If the layout in figure 1 is imaged by a telecentric optical system withmagnification M, then the lateral resolution dx and dy will depend on M and thespatial resolution of the sensor used in the same way as a conventional camera.The lateral and depth resolution define a gauge volume of size dx!dy!dzwithinthe measurement volume, as shown in figure 6. All scattering points within thegauge volume centred at (x, y, z) contribute to the interference signal at position(Mx, My) in the image plane, modulated with frequencies within the rangefKdf=2;fCdf=2. After deformation, the scattering points initially inside the

gauge volume centred at (x, y, z) move to a new position xCu; y; zCw,indicated with a dashed line rectangle in figure 6. w(x, y, z) then represents theout-of-plane component of displacement vector d, which is an average

measurement

volume

O

R

q q

q+q q+q

Figure 7. The measurement volume is defined by the intersection of the illumination beams duringthe tilt scan.

2491Tilt scanning interferometry

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

13/23

displacement of all scattering points inside the intersection of the solid line andthe dashed line rectangles.

From equation (2.3), the total phase change Df introduced in the wavefrontcoming from point F, by a tilt angle Dq introduced in the illumination angle in atime T, is

Dfx; y; zZDf0; 0; 0C2pn0Dq

lx cos qKz1 sin qKzKz1x: 2:21

This phase change introduces CZ jDfj=2p modulation cycles in the interferenceintensity signal. In order to comply with the Shannon sampling condition, thenumber of samples required has to be at least twice the number of cycles:

NfRDf0; 0; 0

pC

2n0Dq

lx cos qKz1 sin qKzKz1x: 2:22

It follows that for a given central illumination angle and tilting range, the

number of samples Nf determines the maximum depth a slice can be within thematerial in order to be able to determine its displacements, assuming thescattering, absorption and laser coherence length do not set a lower limit. Thisdistance is known as the depth range, and is given by

DzzKl

2n0xDqNfK

Df0; y; 0

p

Kz1

sin q

xK1

C

x cos q

x: 2:23

If we think in terms of frequencies, it is easy to realize that the offset frequencyfx; y; 0Zf0; 0; 0Cfxx; y; z ((2.6) and (2.7)) pushes the spectrum towardsthe Nyquist frequency, thus reducing the depth range. In order to reduce this

wasted bandwidth, f(x, y, 0) should be as low as possible, but without interferingwith the cross-interference peaks close to the dc.

The displacement resolution sz

(sometimes called sensitivity of the technique)is decoupled from the depth resolution dzand only depends on the wavelength ofthe laser source and the degree of speckle decorrelation. For out-of-planesensitivity, as is the case in figures 2a and 5, it is typically better than lc/30.

3. Proof-of-principle experiments

(a) Experimental setupIn order to test the proposed technique described in 2, we carried out proof-of-principle experiments, in which the tilt scanning technique was used to measuredisplacements fields within a partially transparent scattering sample. The opticalsetup is shown in figure 5. A collimated continuous wave (CW) beam ofwavelength lZ532 nm and power of approximately 100 mW is steered by mirrorTM mounted on a tilting stage. Left and right illumination beams are obtainedwith the aid of a cube beam splitter CBS and mirrors M1 and M2. TM is tilted bymeans of a piezoelectric leadzirconatetitanate (PZT) actuator controlled by aramp generator RG.

The imaging system consists of an imaging lens L1, field and relay lenses L2and L3, respectively, wedge beam splitter WBS and CMOS high speed camera C(HCC-1000 Vosskuhler). WBS is used with the reflective surface closer to the

P. D. Ruiz and others2492

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from

8/2/2019 Tilt Scanning Interferometry

14/23

object, to avoid multiple reflections of the object beam within the glass thickness.

This is crucial in order to avoid multiple peaks in the spectrum corresponding tothe same depths in the sample. Preliminary experiments with an external BS infront of L1 resulted in multiple reflections of the object beam, causing mixing ofthe spatial information from the interference intensity signal. WBS serves torecombine reference and object beams onto the camera sensor. A smooth on-axisreference wave was used rather than a second speckled wave in order tomaximize the signal-to-noise ratio. The purpose of lenses L2 and L3 is to place theWBS between the imaging lens and the camera sensor. A screen S by the CBSserves for alignment purposes: if the object is replaced by an optical flat alignedwith its normal parallel to the optical axis, then the left and right illumination

beams interfere at S after reflection (transmission) on the optical flat and onto(through) M2, M1 and (CBS). When the left and right illumination beamssubtend the same angle to the optical axis, the fringes are nulled at S. This isimportant because equations (2.14) and (2.15) assume that the angle is the samefor both illumination directions.

Figure 7 shows a close up of the illumination beams in the region surroundingthe object, before and after tilt of the beams by an angle Dq. It can be seen thatthe measurement volume is the intersection of the beams at the beginning andend of the scan range of the tilt angle. Scattering points within this volume giverise to a continuously modulated interference signal throughout the whole tilt

scanning sequence.The test object was a beam manufactured in-house with water clear castingepoxy resin (from CFS Fibreglass Partnership), seeded with a small amount oftitanium oxide white pigment to increase the scattering within the material. Thefront and back surfaces of the beam were polished to reduce superficial scatteringand the back face was painted black to eliminate the scattering produced by thereflected component due to internal reflection. Under a three-point bending test,the beam was loaded with a ball-tipped (6 mm diameter) micrometer against twocylindrical rods (12 mm diameter), as shown in figures 7 and 8. A rough referencesurface R was placed just in front of the object, so as to cover approximately 20%

of the lower portion of the area imaged. As described earlier, this served tocompensate for the shift of the peaks along the horizontal axis x and to allow forcorrection of the nonlinear response of the tilting stage at TM.

field of view

reference surfaceop

tical

axis

x

y

z

loading pin

Figure 8. Schematic of the epoxy resin beam under three-point bending load. In-plane andout-of-plane depth-resolved displacements were measured through the thickness of the beam.

2493Tilt scanning interferometry

Proc. R. Soc. A (2006)

on February 27, 2012rspa.royalsocietypublishing.orgDownloaded from