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Applying laser light, however, gives the scatteredlight a characteristic granular appearance as shown in the image of a speckle pattern inFigure 8.2.. con-It is easily realized that the

8.2 THE SPECKLE EFFECT

In Figure 8.1, light is incident on, and scattered from, a rough surface of height variations

greater than the wavelength λ of the light As is shown in the figure, light is scattered

in all directions These scattered waves interfere and form an interference pattern sisting of dark and bright spots or speckles which are randomly distributed in space Inwhite light illumination, this effect is scarcely observable owing to lack of spatial andtemporal coherence (see Section 3.3) Applying laser light, however, gives the scatteredlight a characteristic granular appearance as shown in the image of a speckle pattern inFigure 8.2

con-It is easily realized that the light field at a specific point in a speckle pattern must be

the sum of a large number N of components representing the light from all points on the

scattering surface The complex amplitude at point in a speckle pattern can therefore bewritten as

Optical Metrology Kjell J G˚asvik

Copyright  2002 John Wiley & Sons, Ltd.

ISBN: 0-470-84300-4

194 SPECKLE METHODS

l

Figure 8.1 Light scattering from a rough surface

Figure 8.2 Photograph of a speckle pattern

SPECKLE SIZE 195

A measure of the contrast in a speckle pattern is the ratio C = σI/ I, where σ1 is thestandard deviation of the intensity given by

σ12 = I2 = (I − I)2 = I2− 2II + I2 = I2 − I2 ( 8.3)

where the brackets denote mean values By using

the screen will receive light contributions from all points on the scattering surface Let usassume that the speckle pattern at P is a superposition of the fringe patterns formed by

light scattered from all point pairs on the surface Any two points separated by a distance l will give rise to fringes of frequency f = l/(λz) (see Section 3.6.1, Equation (3.35)) The fringes of highest spatial frequency fmaxwill be formed by the two edge points, for which

For smaller separations l, there will be a large number of point pairs giving rise to fringes

of the corresponding frequency The number of point pairs separated by l is proportional

z

P

S D

Figure 8.4 Objective speckle formation

196 SPECKLE METHODS

to (D − l) Since the various fringe patterns have random individual phases they will add

incoherently The contribution of each frequency to the total intensity will therefore beproportional to the corresponding number of pairs of scattering points Since this number

is proportional to (D − l), which in turn is proportional to (fmax− f ), the relative number

of fringes versus frequency, i.e the spatial frequency spectrum will be linear, as shown

in Figure 8.5

Figure 8.6 shows the same situation as in Figure 8.4 except that the scattering face now is imaged on to a screen by means of a lens L The calculation of thesize of the resulting so-called subjective speckles is analogous to the calculation ofthe objective speckle size Here the cross-section of the illuminated area has to be

sur-exchanged by the diameter of the imaging lens The subjective speckle size σs therefore

SPECKLE PHOTOGRAPHY 197

where f is the focal length, we get

where m = (b − f )/f is the magnification of the imaging system From this equation we

see that the speckle size increases with decreasing aperture (increasing aperture number).This can be easily verified by stopping down the eye aperture when looking at a specklepattern

Speckle formation in imaging cannot be explained by means of geometrical opticswhich predicts that a point in the object is imaged to a point in the image The field at apoint in the image plane therefore should receive contributions only from the conjugateobject point, thus preventing the interference with light from other points on the objectsurface However, even an ideal lens will not image a point into a point but merely form aintensity distribution (the Airy disc, see Section 4.6) around the geometrical image pointdue to diffraction of the lens aperture This is indicated in Figure 8.6 It is thereforepossible for contributions from various points on the object to interfere so as to form aspeckle pattern in the image plane

8.4 SPECKLE PHOTOGRAPHY

Discussions on the subject of speckle photography can be found, for example, in Burchand Tokarski (1968), Dainty (1975), Erf (1978), Fourney (1978), Hung (1978) and Jonesand Wykes (1989)

8.4.1 The Fourier Fringe Method

Assume that we image a speckle pattern onto a photographic film After developmentthis negative is placed in the object plane of a set-up for optical filtering like that inFigure 4.12 Figure 8.7 shows (Fourney 1978)

(1) a speckle pattern on the negative;

(2) the resulting diffraction pattern (the spatial frequency spectrum) in the xf, yf-plane; and

(3) typical form of the smoothed intensity distribution along the xf-axis

The dark spot in the middle of Figure 8.7(b) is due to the blocking of the strong order component Figure 8.7(c) displays the essential feature discussed in Section 8.2,namely that the imaged speckle pattern contains a continuum of spatial frequencies

zeroth-ranging from zero to fmax= ±1/σs, where σs is the smallest speckle size given fromEquation (8.7) For a circular aperture the frequency distribution will not be exactly linear

as in Figure 8.5, but merely as indicated in Figure 8.7(c)

Now assume that we in the diffraction plane (the filter plane) place a screen with a

hole a distance xffrom the optical axis This situation is illustrated in Figure 8.8, whereFigure 8.8(a, b) shows the spectra before and after the filtering process has taken place.The same spectrum (as in Figure 8.8(b)) would have resulted by filtering out the first side

Figure 8.7 (a) Speckle pattern and its corresponding; (b) diffraction pattern; and (c) intensity

dis-tribution along the x1-axis ((a) and (b) reproduced from Fourney (1978) by permission of Academic Press, New York.)

Figure 8.8 Spectrum of a speckle pattern (a) before and (b) after filtering

As a consequence, we can imagine that a grating given by

is attached to the object surface When the object undergoes an in-plane deformation,the speckle pattern will follow the displacements of the points on the object surface.Consequently, the grating will be phase-modulated and thus can be written as

t2= a[1 + cos 2π(fxx + ψx(x))] ( 8.12)

This is closely analogous to the situation described in Section 7.4 where a model gratingwas attached to the object for measurement of an in-plane deformation by means of moir´etechnique In the same way as described there, we can image the speckle pattern beforeand after the deformation onto the same film negative by double exposure The resulting

transmittance then becomes equal to t1+ t2, the sum of Equations (8.11) and (8.12) Thetwo speckle patterns could possibly also be imaged onto two separate negatives and

subsequently superposed, giving a resultant transmittance equal to t1· t2, but this is moredifficult to achieve By means of optical filtering of the double-exposed negative weget an intensity distribution dependent on the modulation function only (see Section 4.7,

Equation (4.66)) This distribution will be maximum for ψx(x) = n and minimum for

ψx(x) = n + (1/2) and correspond to displacements (cf Equations (7.10) and (7.13))



λ0f

xf

in direct analogy with the moir´e method In deriving Equation (8.13) we have assumed

unit magnification These values therefore must be divided by the magnification m applied

when recording the speckle patterns

The speckle pattern represents gratings of all orientations in the plane of the object (cf.the spectrum in Figure 8.7(b)) Therefore, by placing the hole in the screen in the filter

plane a distance yf along the yf-axis we obtain the modulation function ψy(y) and the

corresponding displacement vy(y) along the y-axis Generally, by placing the hole in the filter plane at a point of coordinates xf, yf, the resulting intensity distribution will give adisplacement equal to

s(x, y)=u2(x) + v2(y) = nλ f



( 1/x )2+ (1/y )2 for maxima (8.14a)

200 SPECKLE METHODS

s(x, y) = (n +1

2)λ0f



( 1/xf)2+ (1/yf)2 for minima (8.14b)

In this case we thus measure the displacement along a direction inclined an angle β to the x-axis given by

In this method we can therefore vary the measuring sensitivity by varying xf and yf, and

a remarkable property of this technique is that the sensitivity can be varied subsequent

to the actual measurement, that is, after the speckle patterns have been recorded Highestsensitivity is obtained by placing the filtering hole at the edge of the spectrum Thefirst-order dark fringe then corresponds to a displacement equal to

smin= 1

2σs= 12

λb

With a magnification m= 1, this gives a sensitivity equal to the laser wavelength tiplied by the aperture number of the imaging lens The sensitivity limit by using laserspeckle photography can therefore be down to about 1µm

mul-Figure 8.9 shows an example of such Fourier fringes obtained by this method (Hung1978) Note that as the filtering hole is moved away from the optical axis, the number offringes increases, i.e the sensitivity is increased

Figure 8.9 Fringe patterns depicting the horizontal and vertical displacements of a cantilever beam obtained from the various filtering positions in the Fourier filtering plane (Reproduced from Hung 1978 by permission of Academic Press, New York.)

SPECKLE PHOTOGRAPHY 2018.4.2 The Young Fringe Method

On photographing a speckle pattern, each speckle will form a pointlike blackening on thefilm When the object undergoes an in-plane deformation, the speckle pattern will followthis deformation A double-exposed negative of two speckle patterns resulting from adeformation therefore will consist of identical point pairs separated by a distance equal

to the deformation times the magnification of the imaging system

Assume that we illuminate this double-exposed negative with an unexpanded laserbeam When the beam covers one pair of identical points, they will act in the same way

as the two holes P1 and P2 in the screen of the Young’s interferometer (see Figure 3.13,

Section 3.6.1) On a screen at a distance z from the negative we therefore will observe

interference fringes which are parallel and equidistant and with a direction perpendicular

to the line joining P1 and P2, i.e perpendicular to the displacement

The situation is sketched in Figure 8.10 In Section 3.6.1 we found that the distancebetween two adjacent fringes in this pattern is equal to

d= λz

where D is the distance between P1 and P2 If the displacement on the object is equal

to s the separation of the corresponding speckle points on the negative is equal to m · s where m is the magnification of the camera Put into Equation (8.17) this gives, for the

object deformation,

s= λz

In deriving the equations for the Young fringe pattern in Section 3.6.1 we used the

approx-imation sin θ = tan θ Without this approximation Equation (8.18) becomes

By measuring the fringe separation d we can therefore find the object displacement at the

point of the laser beam incidence using Equation (8.18) Better accuracy is obtained by

measuring the distance dn covered by n fringes on the screen We then have d = dn/n,

z

d Negative

Laser

Screen

Figure 8.10 Young fringe formation

202 SPECKLE METHODS

which gives

s = nλz

To obtain such a Young fringe pattern, the identical pairs of speckles must be separated

by a distance which is at least equal to one half of the speckle size, that is 1/2σs This isthe same sensitivity limit as found in Equation (8.16)

By placing the film negative into a slide, movable in both the horizontal and verticaldirections, measurement on different points on the film, i.e the object surface, is performed

Figure 8.11 Young fringes at different points in a plate under tension in a miniature rig

SPECKLE CORRELATION 203

quite easily and quickly This is, however, in principle a pointwise measurement in contrast

to the Fourier fringe method, which is a full field measurement On the other hand, theYoung fringe method gives better accuracy and the fringes are more easily obtained Ifthe Young’s fringes do not appear from a double exposed specklegram it is, under normalcircumstances, also impossible to obtain the Fourier fringes

Figure 8.11 shows an example of the results obtained by this method The object is ametal plate under tension in a miniature rig

8.5 SPECKLE CORRELATION

As we have seen, speckle metrology is mainly concerned with the measurement of plane deformations of objects When a laser-illuminated diffuse surface undergoes adisplacement and/or deformation, the speckles in the scattered field or in its image show acorresponding displacement This displacement can be represented by the peak position of

in-the cross-correlation function c I X between the intensity distributions I1(x, y) and I2(x, y)

(specklegram 1 and 2) of the speckle patterns before and after the object displacement.Physically, the correlation process can be visualized as the sliding of specklegram 1 over

specklegram 2 and an assessment of the similarity between I1 and I2 for each value of

the lag Mathematically, the cross-correlation function c I X is defined by multiplying theintensity at each point on specklegram 1 by the intensity at the point on specklegram

2 displaced from it by the lag distance (components x = x2− x1 and y = y2− y1),averaging over the whole area, and repeating for different values of the lag:

c I X (x1, y1; x2, y2) = I1(x1, y1)I2(x2, y2) ( 8.21)

where . denotes the spatial averaging.

Figure 8.12(b) shows speckle patterns recorded by an electronic camera before andafter in-plane translation of a piece of paper The two-dimensional cross-correlation was

computed by a digital computer For comparison, the autocorrelation (I1= I2) of thepattern before translation is shown in Figure 8.12(a) The peak of the autocorrelation

is always located at zero The peak of the cross-correlation corresponds to the speckledisplacement and the decrease in peak height is associated with change in the structure,so-called decorrelation

To analyse the laser speckle phenomenon further, we have to specify the statistics ofthe amplitudes of the speckle field When assuming Gaussian statistics (see Section 8.2)

it is an accepted fact that the autocorrelation (Goodman 1975)

c I (x1, y1; x2, y2) = I (x1, y1)I (x2, y2)  = I (x1, y1) I (x2, y2)  + |c u (x1, y1; x2, y2)|2

( 8.22) Here, I (x, y) = |u(x, y)|2 is the intensity and

c u (x1, y1; x2, y2) = u(x1, y1)u∗(x2, y2) ( 8.23)

is the autocorrelation function of the fields, also referred to as the mutual intensity

To calculate the field u(x, y) in the xy-plane resulting from free space propagation from an illuminated rough surface in the ξ η-plane, where the field is given by u (ξ, η),

ory and applications of speckle displacement and decorrelation,’ in Sirohi, R.S., Speckle Metrology.

Reproduced by permission of Marcel Dekker, Inc., New York

we apply the Fresnel approximation (writing out the x, ξ -dependence only):

2

Now we assume the mutual intensity in the ξ η-plane to be given as

c u0(ξ1, ξ2) ∝ P (ξ1)P∗(ξ2)δ(ξ1− ξ2) ( 8.26) where P (ξ ) is the amplitude of the incident field This means that we assume zero correlation except when ξ1 = ξ2 Equation (8.26) put into Equation (8.25) gives