Displacement Measurements Using Double-Exposure Speckle Photography of Small-Angle Scattered Light

Synopsis

1)oulile-exposure speckle pattern:, created Iiy s c a t t e d light were used t o study tensile detormation 01' isotactic polypropylene film. T h e method makes po~s ih le a quantitative investigation 0 1 ' the def'ormation process (relative magnitude and orienttit ion of'the displacement vector, Poisson's ratio p ) iiy means o f a simple optical analysis of'.;tep\vihe tlef'ormations covering the desired range. T h e displacement vector and Poisson's ratio have lieen determined with an accuuracy of' f 5S in the range of' small def'ormations ( u p to i8(I relative detormation): and the role of' stress relaxation has heen examined. 1,imitat ions of the rnethod. such as the range of' measuralile displacements. elf'ect of' prestressing, and stability of' the system. etc.. are discussed. Among advantages of' the method are comparative simplicity and accuracy. tind the possil)ility ot its application to other systems studied by the small-angle light scattering method.

INTRODUCTION

If laser light is used as the source in small-angle light scattering (SALS), scattering patterns have a random fine structure called "speckling." The sta- tistical properties of speckle patterns depend on the structure of the object studied and on the properties and arrangement of optical elements (source and its coherence, apertures, lenses, etc. 1. Here, we shall examine only "laser speckle" connected with scattering object, because in S A I S one can usually dispense with an image-forming system.

Thus far, speckling of scattering patterns has been regarded as an undesired effect which leads to a loss of pattern quality. Recently, methods have been proposed to characterize and utilize speckle patterns caused by direct irradiation of an object (coherent light is scattered by a rough surface).' Further, analysis of speckle patterns created by scattered light has heen used in the investigation of the properties of a transparent medium.' It has been shown that the cor- related character of scattered-light speckle patterns may be utilized in a quan- titative study of mechanical displacements of morphological structures examined by SALS. Up until now, the method has heen employed to determine the di- rection and magnitude of relative displacements and of Poisson's ratio p in de- formation of a spherulitic film of isotactic polypropylene at very low deformations (tensile deformation several percent). To deal with larger deformat ions the method has been extended by using an optical analysis o f double-exposed scat- tered-light speckle patterns i n small consecutive deformations (deformation steps) covering the range under investigation. The use of the method in a stress relaxation study and some of' its limitations are also discussed.

,Journal of' Polymer Science: Polymer Physics Edition, \ '()I . 18. 265-275 ( 1980) ' C 1980 ,John Wiley & Sons, Inc. oogx- I m / 8 o / o o t8-i)m.Yio I .oo

266 HOLOUBEK AND SEDLACEK

X /

Fig. 1. Scheme of speckle-pattern detection. The sample in situated in the object plane P; scat- tered light is recorded in a far field in plane X.

THEORETICAL CONSIDERATIONS

In the current method? scattering patterns at 90° to the direction of the in- cident beam are used without regard to the morphological aspects of the problem. Our study, however, is based on the assumption that changes in the internal morphology reflected in changes of SALS patterns are also reflected in changes in speckling of these patterns. In the arrangement employed for SALS (Fig. 1) a photographic exposure is obtained with the system in some state (e.g., the Hv pattern of an undeformed film containing spherulites), after which the structure is altered (e.g., by a slight deformation of the sample) , and exposure of the changed Hv pattern, including speckling, is recorded on the same photograph. In this way, a doubly exposed Hv pattern is obtained. In principle, two methods can be employed to evaluate information provided by the double-exposure speckle patterns, namely, a whole-field and a pointwise Fourier transform.* In this work pointwise filtering has been used, because it has some advantages. These consist mainly in a more exact determination of the distance between fringes, the formation of sharper fringes, and also in the advantage of lensless processing (avoiding lens aberration). The arrangement used in the pointwise filtering (Fig. 2) comprises a source (laser), the speckle pattern to be analyzed, a filtering mask allowing selection of a suitable small area of the pattern, and a

Fig. 2. Schematic arrangement for pointwise filtering. The double-exposure speckle pattern is in plane X, fringes are recorded in plane 2.

DISP1,ACEMENT MEASUREMENTS 267

detection system (plate or f'ilm) placed a t a distance L from the speckle pattern plane. The intensity of light in the recording plane 2 is given by:<

(1 )

where d1,d.~ are coordinates of the displacement vector in the specklegram plane X, and s 1,s2 are coordinates a t the diffraction receiving plane 2'. The function I 1 is the intensity distribution at the receiving plane due to diffraction from a single speckle pattern. Equation (1) shows that the double-exposure speckle pattern, with a change in speckling between exposures, leads to a modulation of the function II (sI ,s2) by cosine-square fringes. The sequence of dark fringes satisfies the condit,ion

( 2 )

The fringes are an assembly of intercepts perpendicular to the displacement vector d, uniformly spaced with a density proportional to the magnitude of the displacement vector Id/. It is known that the smallest measurable displacement in the specklegram plane is d,ni,, = c~/1.2, where 0 is the grain size.

The relative displacement is mapped into the object plane using t,he rela-

I(s1,s.)) = 4 cos"k(s1d, + * s ~ d ~ ) l ' l ] I l ( s l , . s . J

s d = ( n + '/2)AL, n = O,f 1,f 2,. . .

tan(d0) * d0 = d , / H ,

H , = (p : + A ? ) ] / ? ; I,, = ( s f ( + x i l ) l l 2

The meaning of symbols can he seen in Figure 1. Poisson's ratio p was calculated by means of5

P = ( p r / r ) l / ( h - / r ) , (4)

and ofe

p = '/> ( A - $ ) / ( A - l), $ = A l A

where X I and XI are the extension ratios, respectively, parallel and perpendicular to the extension direction.

EXPERIMENTAL

After obtaining a double-exposure speckle pattern, one has to visualize Young's fringes by Fourier transformation of the speckle pattern. The two stages of the experiment were carried out as follows:

(a) The arrangement used to produce double-exposure speckle patterns is the same as that used in the photographic SALS method; (Fig. 1). The light source was a Spectra Physics He-Ne laser. Model 124, in the single-mode regime. The sample was fixed in a holder of our design, permitting a uniaxial deformation of the polymer film (symmetrical tensile stress, i.e., both clamps mobile). The polarizers were in a parallel and crossed arrangement ( V i . and H L ~ scattering patterns). The pattern was recorded on negative photographic film (Polaroid 105 P/N film) with resolution of 160-180 lines/mm. Some double-exposure patterns were also recorded on a material with higher resolution (Scientia Agfa

268 HOLOUBEK AND SEDLACEK

Fig. 3. Single speckle pattern in plane X with indicated positions of filtering area A1-A15.

Gevaert 8375) in order to ascertain the possible loss of information due to the comparatively low resolution of the Polaroid film.

The sample was prepared from a film of isotactic polypropylene, 160 pm thick, with a spherulitic structure obtained by remelting and self-seeding. The pho- tographic exposures were made on a vibration-isolated table used for holography. The time of double-exposure was 2 X 11125 sec.

(b) For the Fourier transformation the negative of the double-exposure pat- tern was processed by pointwise transform, as indicated in Figure 2. For this purpose, a mask with circular apertures situated at various points of the pattern was prepared. The relative size and position of the apertures are shown in Figure 3. If a narrow beam passes through the small area of the negative limited by the mask, one observes, at a distance L, Young's fringes with density and orientation corresponding to eq. (2).

RESULTS AND DISCUSSION

The deformation experiments are listed in Table I. A relative tensile defor- mation of 1.5% (determined by means of a comparator from the clamp dis- placements) was chosen as the partial deformation step for a pair of exposures on the basis of preliminary tests. Double-exposure patterns (usually with a change of tensile deformation between the exposures, see Table I) on Polaroid 105 PIN film are denoted by N , those made on Scientia 8375 plates are denoted by HN. Patterns recorded during stress relaxation are indicated by R. Inter-

DISPLACEMENT MEASUREMENTS 269

TABLE I Deformation Experiments

Initial Statea 0 1 2 3 4 7 11 14 16 18 Final statea 1 2 3 4 5 8 12 14 16 18 Designation N1 HN2 N3 N4 N5 HN8 N12 N14R N16R N18R

a Each unit step denotes 1.5% macroscopic deformation.

pretation of initially observed displacements ( N I ) was complicated by difficulties in the definition of the initial undeformed state. The displacements shown in Figure 4 are due superpositions of the micromotion of scattering centers and of macromotion of the sample (motion of the sample as a whole owing to buckling, sag-deflection of the sample, etc). These contributions cannot be discerned without employing special procedures, such as deblurred speckle photography or polychromatic speckle patterns. We therefore restrict ourselves to noting the characteristics typical of the initial step double-exposure. These are (a) discrepancies between the directions of displacements and the direction of clamp displacement, (b) an increase of displacements toward the center of the pattern (toward the optical axis), and (c) a maximum displacement that does not occur in the direction of deformation. In the initial double exposure at least one of these features is always present; and, as a rule, all three can be observed. In the second deformation step ( H N 2 ) the displacement exhibited a decreased effect of macromotion; unfortunately, however, no speckle pattern with sufficient contrast was obtained at positions A l , A2, A13, A14 owing to inadequate sen- sitivity of the photographic emulsion. Similarly, in several cases the evaluation does not yield discernible fringes. Such cases are denoted by the horizontal dashes outside empty circles in schematic diagrams (Figures 4 and 6). Further deformation experiments N3, N4, N5 (Fig. 5) exhibit similar dependences of the displacement vector and illustrate possibilities offered by the method, namely, comparatively accurate determination of the directional dependence of the displacement vector of the reference point for small deformation, and

I 75'

I

la) (b 1 Fig. 4. Schematic diagrams of double-exposure scattering patterns: experiments N1, HN2.

Displacements in the object plane P are given in percent, the orientation of the central fringe is in- dicated with given angular orientation of fringes at points along directions + = 0", 30°, 45", 60°, and 90" with respect to the deformation axis; (a) N1, (b) HN2.

270 HOLOUBEK AND SEDLACEK

Fig. 5. Experiment N3, actual appearance of fringes in plane 2. Displacements in the object plane P are given in percent, angular positions of fringes in degrees.

determination of Poisson’s ratio p. Values of p determined using Eqs. (4) and (5) are summarized in Table 11. Calculations were carried out using both a simple average of displacement and a weighted average assigning double weight to fil- tering positions A2 and A14. The weighting seemed justified since in these positions a large number of well defined fringes were formed. The accuracy in determination of p was estimated as f 5% (the standard deviation of the weighted average was less than f 0.03).

In experiment N 1 2 we used a Vv double-exposure speckle pattern (with the polarizer perpendicular to the deformation axis). The optical processing yields results in a very good agreement with those obtained for smaller deformations (cf. Table I1 and Fig. 6). This finding was also confirmed by relaxation experi- ments, in which a Vv double exposure was used. In experiments N14R, N16R, and N18R (Fig. 6)’ stress relaxation was examined. The first exposure was made immediately after the deformation step; the second exposure followed after 10 min. The following conclusions maybe drawn from the relaxation experiments: (a) The high degree of correlation of speckles is preserved over a rather wide range of deformations (experiment N18R corresponds to a relative deformation of

TABLE I1 Poisson’s Ratio

Experiment N3 N4 N 5 HN8 N12 Equation used

0.422 0.431 0.441 0.445 0.445 5 weighted average 0.429 0.424 0.443 ... 0.445 5 average 0.424 0.418 0.437 0.440 0.442 4 average 0.418 0.425 0.435 ... 0.443 4 weighted average

DISPLACEMENT MEASUREMENTS 271

0.

0.

0.

0.25 0 2 L 023

(b)

0.36 0.32 053

(c) (d) Fig. 6. Experiments N12, N14R, N16R, and N18R, symbols as in Figure 4; (a) N12, (b) N14R,

(c) N16R, (d) N18R.

about 27%). (b) Up to the tenth deformation step (i.e., 15% deformation), the method detects no displacements under the experimental conditions used (re- laxation experiments 5 - 5,6 - 6, and 9 - 9 showed no measureable displace- ments). (c) With increasing total deformation, larger displacements take place during relaxation. (d) Poisson’s ratio does not change in the relaxation exper- iments, and the measured displacements have the same orientation and relative magnitude as during the deformation tests. (e) During relaxation, as in tlie high deformation experiments N4, N5, etc., a slight deviation of direction of the displacement vector from the theoretical value is observed (cf. Table 111).

Relative values of the displacement vector in the object plane given in Table I11 represent averages of the displacements for a given orientation. Weighted average values (in parentheses) were obtained by assigning positions A2, A5, A8, A l l , and A14 double weight. Theoretical values given in Table I11 were deter-

TA

BL

E I

11

Com

paris

on o

f The

oret

ical

(T

) and

Exp

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enta

l (E

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plac

emen

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ar O

rien

tatio

n of

Fri

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Posi

tions

on

scat

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tter

n Ex

peri

men

t A

l-A

3 A

4-A

6 A

7-A

9

A 10

-A 1

2 A

13-A

15

T

0.53

5 E

0.54

(0.

54)a

di

spla

cem

net

(%)

N3

T

90

82

orie

ntat

ion

(deg

)

0.79

0.

75 (

0.76

)

36

39

0.98

1.

01 (1

.02)

23

23

1.14

1.

09 (1

.10)

14

12

1.28

1.

28 (1

.28)

0 02

x 0

r 2

1.26

1.

47

1.65

2

1.41

(1.4

1)

1.67

(1.6

5)

8 3

T

0.70

1.

02

E T

90

36

23

14

0

0.70

(0.7

0)

0.93

(0.9

4)

1.26

(1.2

5)

disp

lace

men

t (70

)

N4

> 88

44

25

14

02

or

ient

atio

n (d

eg)

T

0.69

0.

70 (

0.69

) di

spla

cem

ent (

%)

N5

1.00

0.

92 (0

.92)

1.

23

1.21

(1.2

2)

1.42

1.

33 (1

.32)

1.

59

1.59

(1.5

9)

T

90

89

orie

ntat

ion

(deg

)

T 0.

67

0.67

di

spla

cem

ent

(%)

37

42

0.95

0.

90 (

0.88

)

24

26

1.17

1.

19 (1

.17)

14

15

1.36

1.

27

0 01

1.52

1.

52

HN

S T

90

37

24

14

0 90

49

29

16

0 or

ient

atio

n (d

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a W

eigh

ted

aver

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alue

s in

pare

nthe

sis.

DISPLACEMENT MEASUREMENTS 273

mined from relations for the motion of a reference point in tensile deformation with the values of Poisson's ratio p (weighted average) from Table I1 for exper- iments N3, N4, N 5 , and HN8. The agreement between theoretical and exper- imental values is very good. Small deviations in the assumed orientation of the displacement vector (especially in positions A4-A6) are probably due to steric obstructions in the orientation of the comparatively closely packed system.

The accuracy of determination of fringe orientation was f 2'. We estimate for the limit of accuracy of the method a relative error of f 3 % in the determi- nation of displacement (corresponding to 1/,%,) fringe). Such accuracy could be achieved by choosing an optimal filtering area with many well defined fringes, a t least in cases of well developed speckling.

It is known that the optimal sensitivity of the method is limited to measure- ments of displacements larger than the diameter of an individual speckle (both values must be considered in the double-exposure speckle plane). This repre- sents a low bound for the magnitude of displacements that can be investigated. In our case the speckle grain size was about 20 pm, which in the object space corresponds to a relative displacement of about 0.2%) in the optimal filtering areas A 2 , A5, - - . The choice of the values of the deformation step is not too critical, a t least within limits, because one usually has the additional choice of a suitable filtering area. The maximal measurable displacement is limited by the corre- lation of speckles. Destruction of the correlation is due to the nonuniform mo- tion of scattering centers in the irradiated volume, relaxation, and phenomena connected with the macromotion of the sample, as well as to some other effects. We have shown in our experiments that the measurable correlation survives to a deformation of about 27% (experiment N18R). It has been found, in practice, that with an increasing deformation step between pairs of exposures the corre- lation of speckles is lost more quickly.

Incipient neck formation is reflected in a nonsymmetrical distribution of displacements in four quadrants formed by the axes of the polarizers (in the case of H v scattering), or by the axis of the polarizer and that of deformation (in the case of VV scattering). This asymmetry can be seen in the double-exposure speckle pattern under an optical microscope; usually by processing the double- exposed pattern one can obtain the values of the displacement vector. A check of the sample with an optical microscope confirmed the existence of an incipient neck in the area, indicated by a permanent change in birefringence in the band passing through the originally irradiated volume. In some cases the asymmetry in the double-exposure speckle pattern can be detected visually; in such case, the displacement values cannot be obtained by using pointwise filtering (cf. Fig. 7); the neck can then be seen microscopically.

The use of polaroids in recording the double-exposed pattern suppresses the influence of surface effects (especially in the case of H L , ) and leads to well "de- veloped" speckle patterns (with high contrast). As regards stability require- ments, mechanical stability should be quaranteed so that no vibrations or mo- tions comparable with the speckle size occur during the exposures. The re- quirement concerning coherence of the laser used is higher in the double-exposure record, when it is necessary that the coherence length of the laser ought he larger than the maximum irradiated volume. This usually causes no problem with continuous lasers. The photographic emulsion used in speckle recording must allow resolution of individual speckles.

274 HOLOUBEK AND SEDLACEK

I)ISPI,ACEMEN‘I’ MEASUREMENTS “75

One disadvantage of the method is that it does not provide the absolute di- rection of displacement (i.e., sign of the displacement). Even though it is pos- sible, in order to overcome this disadvantage, to carry out a known translation of the sample or the photographic plate between exposures, one must estimate in advance the type of the displacement under study. In the general case of unkown displacements this procedure also has some problems.

In this paper we have reported new findings on the deterministic character of speckles created by SALS. We confirm the correlation of speckles during the deformation and shows how this may be employed in quantitative character- ization of the deformation process. The role of relaxation is documented, and the possibility of its study indicated. Among advantages of the met hod one should rank its comparative simplicity and accuracy, and also the possibility of its application to other systems investigated by SALS using laser light.

References

1. M. May and M. FranCon. J . O p t . Soc. Am, , 66, 127.5 (1976). 2. F. P. Chiang, in ?‘/I(, Enginrc~ring 1 ‘sw o/ Co/ic,rcnt Optics . E. R. Hot)ertson. Ed. Cambridge

3. -1. Holouhek and .J. RGiek, Opt . Actn. 26, 43 (1979). 4. R. P. Khetan and F. P. Chiang. Appl. Opt . , 15,2206 (1976). .5. N. H. Ladizesky and I. M. Ward, J . Mncromol. Sci. Phyv.. R.?, 661 (1971). 6. R. S. Moore, J . f’olym. Sci., A-2,711 (1967). 7. H. S. Stein. in Structure and Propc~rt ic~s ( J / f’oljmc~r Films, R. W. I.enz and K. S. Stein. Kds.,

Lrniversity, Cambridge, England, 1976, p. 249.

I’lenum, New York. 1973.

Received September 29, 1978 Revised ,June 20, 1979 Accepted .July 9, 1979