Polychromatic speckle technique for three dimensional non-destructive photoelasticimetry

Volume 21, number 2

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POLYCHROMATIC

SPECKLE TECHNIQUE FOR.

THREE DIMENSIONAL NON-DESTRUCTIVE C. FROEHLY

May 1977

PHOTOELASTICIMETRY

and R. DESAILLY

Laboratoire de Mkanique des Solides, Equipe de Recherche asso&e 40, Avenue du Recteur Pineau, 86022 Poitiers Cedex, France

au C.N.R.S.,

Received 7 February 1977 Isoclinic and isochromatic fringes of slices optically isolated in birefringent models are visualized using spectroscopic analtime impulse response and spectral transfer function of ran-

ysis of speckle patterns in polychromatic radiation. Concepts of dom birefringent linear optical pupils are involved in this work.

Introduction. Non destructive three dimensional photoelasticimetry was made possible many years ago [l-3] by using analysis of polarization properties of light diffused inside photoelastic materials. The refined and sensitive techniques now available work by point determination of local birefringence in models under test; they lead to very accurate measurements. Situations exist - mainly in industrial evaluation of mechanical structures - where such a good precision is not needed, but where global information on stress distribution could be wished. “Stress freezing” followed by mechanical slicing of the model can be applied in this aim. This is obviously a destructive so.lution, which also causes difficulties due to the low compressibility of materials used in this technique. Recently, our laboratory investigated a new way [4,5] : non destructive observation of two-dimensional fields of isochromatic and isoclinic fringes of photoelastic slices inside three dimensional models was demonstrated with the help of “speckle techniques” in spatially coherent radiation. The contrast of the fringe systems was rather poor, due to the low signal-to-noise ratio of the channel recording and processing this information. This might give rise to misleading interpretations of the observed patterns. We present an improvement of this method which strongly increases the signal to-noise-ratio by taking advantage of the supplementary amount of information which can be carried by the “spectral variable”

258

(time frequency

or wavenumber).

Basic ideas. A slice of birefringent

material under test will be singled out (fig. 1) between two plane light sheets (XI) and (Z2) issued from one polychromatic laser source (dye laser). A spectroscopic system (Sp) projects a spectrally resolved image of the illuminated region onto the observation plane (7r). This image consists of the superposition of scattering regions Pl and P2. The superposition arises in amplitude or in energy according to the degree of mutual coherence between vibrations diffracted from PI and P2. Full coherence (or incoherence) corresponds to the situation where the states of polarisation of these vibrations are identical (or orthogonal) [6]. In case of full coherence, amplitude superposition of spectra displayed in plane rr gives rise to spectral interference effects, that is to “channelled spectra”. In case of full incoherence, the power spectra of light diffracted at points PI and P2 overlap without to interfere: there is no channelled structure in these regions. The situation obviously depends on the average birefringence of the slice bounded by planes (21) and (Z2) along the path PI P2 : we showed in previous papers [4,5] that lines of maxima and minima of mutual coherence are exactly the isoclinic and isochromatic fringes we would observe if the slice was mechanically cut and placed between parallel polarizers. Photographs a) and b), fig. 2, seem like speckle patterns; they exhibit - not, in the case of b) - a fine

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Fig. 1. Experimental

set up to record

the spectrally

resolved

images superposition.

“channelled” substructure. They were selected in the image plane (rr) of the spectroscope (Sp) in regions of maximal, or minimal, coherence degree between vibrations issuing from planes (X1) and (X2). It is possible to deduce isoclinic and isochromatic lines of the slice from direct microscopic observation of the whole image recorded in plane (n). But the operation is quite uneasy. It is simpler to use optical filtering for revealing the locus of maximal or minimal contrast of the channelled pattern. Fourier optics description. The photographs in fig. 2 demonstrate that the diffusing model alters the spectral properties of the polychromatic radiation it transmits. It acts as a linear filter in the space of time frequencies - or wavenumbers. The calculations will be performed using concepts [7] of “time impulse response” and “time frequencies transfer” of optical linear pupils in polychromatic light. We suppose the pupil, here the photoelastic model, to be excited by a Dirac impulse, we then calculate its time coherent - although non-monochromatic* “time impulse response”, from which we deduce the time frequencies transfer by Fourier transformation. At first, for the sake of clarity, we calculate the * Let us recall that time coherence Fig. 2. a) Channelled speckle pattern corresponding to region of mutual coherence. b) Unchannelled speckle pattern corresponding to region of mutual incoherence.

of light does not refer to quasimonochromaticity at all, but rather to the deterministic behaviour of modulus and phase functions m(t), $J@) in the complex amplitude a(t) = m(t) expb 0(t)}; time incoherent sources being described statistically by autocorrelation function and power spectrum of the radiation.

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time impulse response and modulation transfer function of a transparent non-birefringent non-dispersive scattering volume along a direction normal to the illumination beam. Only the next step will consider the effects of dispersive and anisotropic properties of the refractive index. As entrance signal, two identical narrow pulses 6 1 and 6 2 (fig. 3) split in an interferometric device (I) (fig. 3) propagate along two parallel beams (B1) and (B2) respectively. The light vibration diffracted by region (R) along direction (0) consists of a sequence (S) of two delayed random pulses, each of them being spread over the finite optical depth e of beams (B1) and (B2). The delay of the second pulse with respect to the first one equals the separation d of the beams. Each pulse has zero average value and same r.m.s. of the complex amplitude - that is same optical energy - in case of diffusing centers homogeneously distributed. This pulse shape may be understood from intuitive considerations, or deduced from a theorem (7) after which “the time impulse response of any linearly diffracting optical pupil is the first derivative of the wave surface transmitted by the pupil along the observation direction”. Thus, amplitude a(Z) in the pulses sequence may be considered as the product of a stationary random function r(Z) by a deterministic modulation h(Z): a(Z) = r(Z) * h(Z),

(1)

with

May 1977 iLl2

lim +J L--t-

r(Z) * dz = 0,

(2)

-L/2

h(Z) = lrect (Z/u)1 @ I6(Z - d/2) t F(Z t d/2)1;

(3)

symbols 8 and 6 denote the convolution operation and the Dirac distribution, respectively; rect. (Z/u) = 1 if -a/2
($FL!&y’.2 cos nd

&

(4)

u’

Expression (5) describes a spectral random structure, whose autocorrelation function C(u’) = .4(u) * A(U) is easily found proportional to the deterministic part of eq. (5) on condition that the random spectrum R(u’) is an uncorrelated stochastic process: c(u’)=Ks*

cosrrdu’.

Convoluting the random distribution R(u’) with the deterministic function C(u’) generates a speckle-like pattern in space of variable u. The “correlation length” of this speckle is Au’ = 1/a and the pseudo-periodicity of its high frequency spectral modulation (channelled spectral speckle) is 60’ = l/d. In birefringent materials, the light scattered by some regious consists or two successive pulses of the sequence S with orthogonal light forms, in this case they are mutually incoherent and they cannot be added in amplitude. Expression (3) may be rewritten in the two dimensional space of the light forms i, k h(Z) = rect (Z/u) @ IS(Z - d/2) i + 6(Z + d/2&l.

(7)

Hence

A(u)=aR(u’)

+R(u’) Fig. 3. Schematic figure showing two random pulses scattered by a transparent medium in case of a narrow impulse excitation.

==

c3+-$$e -jsaa'

sin nau’ +jndo 0 7 e

c3

k

pl(u’)i+R2(u’)kl,

i

(8) (9)

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where R 1 (u’) and R 2(u’) are two uncorrelated

stochastic process. In this case, the spectral plane (n) is illuminated simultaneously by two independent speckle patterns S,(a) and S,(a) with orthogonal polarizations i and k, with same correlation length Au’ = 1/a but no high frequency spectral channelling. The total energy at point u’ in the spectral plane is given in each situation by:

E(u) = K 4(o)12

(IO)

either Q-

2 sin 7rau’ * cos ndu’ 7rau’

(coherer It superposition)

(I I)

or 2

I I

t R2(u’)

(incoherent

superposition)

8

(12)

(01,/3, are photometric constants). Intermediate situations could be analyzed by introducing the scalar product y = i k of the light forms, which would play the role of a partial degree of mutual coherence. l

May 1977

In all the foregoing sections, we considered a nondispersive scattering medium. Now, what would be the effect of dispersion of the phenomena we described above? Dispersion causes relative phase retardation of spectral components: instead of s(t) arriving at P, and P,, we get “chirped” impulses. The amount of chirping along path A1P1P2 is obviously larger than along path .A2P2: dispersive path PIP2 is responsible for an approximately quadratic phase difference between the spectra of vibrations at P, and P2. The interference fringes between these spectra in spectral plane (n) are thus no more equidistant: the spectral speckle is channelled by non-periodic modulation. This effect is so much less important as the light radiation has narrower spectral bandwidth. Spectrograms processing by optical filtering. As mentioned earlier the recorded spectral speckle can be observed in an optical filtering device which gives an image of isoclinic and isochromatic lines without requiring direct microscopic observation of the fine structure of the speckle. The experimental set-up (fig. 4) is a classical double diffraction system with spatial filter consisting of an opaque mask with two parallel slits, the spacing of which is matched exactly to spatial frequencies of the modulating cosine function (of relation (5)): light diffracted by channelled regions may be transmitted through these slits and take part to formation of the

Fig. 4. Spatial filtering set-up to exhibit the regions of mutual coherence (i.e. isoclinics and isochromatics).

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May 1977

Further developments of our study mainly concern: _ extensions to configurations reproducing other various working conditions: slice analysis between crossed polarizers, or in circularly polarized light etc. - quantitative evaluations of possible performances of the method (resolvance, signal to noise ratio ...) - application to three dimensional dynamical problems by sampling techniques in cases of reproducible phenomena, by real time processing of the spectrally resolved slice image in other cases.

Fig. 5. Fringes obtained for a part of slice singled out near the loading point in a short beam in flexure.

filtered image; light diffracted by unchannelled regions cannot. Channelled regions are bright, unchannelled, dark, in the filtered image of the slice (fig. 5). Conclusion. We demonstrated a new possibility to perform non-destructive global three-dimensional photo. elasticimetry by observation of mutual coherence variations between polychromatic fields diffracted at the boundaries of a photoelastic slice (optically singled out in the stressed model). The informations about the birefringence field under test are exactly of the same nature as if the slice was analyzed between two parallel polarizers. Well-contrasted isoclinic and isochromatic fringes systems can be obtained without immersion of models in liquids performing refractive index compensation. The method is expected to be successful in solving industrial problems as it leads to results quite similar to these of usual two-dimensional photoelasticimetry without involving sophisticated experiments. The only practical limitation from the industrial point of view is the need for rather expensive dye lasers as polychromatic light sources.

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It is a pleasure for us to thank Prof. A. Lagarde which communicated us his large experience and his enthusiasm about the three-dimensional photoelastic problem. We are indebted to Dr. Minard (Spectra Physics France) and to Dr. Del Homme (Laboratoire de Physique - Poitiers) for their kind loan of the light sources - Argon Laser pumped dye cell - and the excellent diffraction grating we used in the experiments here reported. We also wish to thank M. Masse for his very efficient assistance during the experiments.

References 111 A. Robert et E. Guillemet,

Rev. Francaise de Mecanique, (1963) no. 5-6, p. 147-157. Exp. Mech (1974) 317. 121 J.G. GrossPetersen, [31 J. Brillaud et A. Lagarde, C.R. Acad. SC. Paris, 281 (1975) 19 (1976) 61. [41 R. Desailly, Opt. Commun. des proprietes des [51 R. Desailly et A. Lagarde, Application champs de granularite a la photoelasticimdtrie tridimensionnelle, C.R. Acad. SC. Paris, a paraitre. [61 L. Mandel, E. Wolf, Rev. Mod. Phys. 37 (1965) 231. 171 C. Froehly, A. Lacourt, J.Ch. Vienot, Mouv. Rev. Opt. Appl. 4 (1973) 183.

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