holographic fringe-contrast interferometry

Optics and Lasers in Engineering 9 (1988) 121-135

Holographic Fringe-Contrast Interferometry

Marek J. Matczak

Institute of Physics, University of Szczecin, ul. Wielkopolska 15, 70-415 Szczecin, Poland

(Received 19 June 1987; revised version received and accepted 10 February 1988)

ABSTRACT

A brief theoretical background and physical principles of holographic fringe-contrast interferometry are given. A practical procedure of fringe-contrast measurement is described. An example of the interpretation process for evaluation of the displacement field, with illustrative experiments, is presented. Sensitivity and accuracy of the technique is also discussed.

1 INTRODUCTION

The name Holographic Fringe-Contrast Znterferometry (HFCI) is pro- posed for measuring techniques and interpretation methods which are based on information contained in the fringe-visibility distribution in holographic interference images.

Fringe visibility was first utilized in the fringe-localization method14 of holographic interferometry for investigation of rigid-body motions and homogeneous deformations. Unfortunately, this method has a limited range of applications and the measuring process with this method is time-consuming and difficult from a technical point of view.

The analysis of the information content of the fringe-visibility distribution in holographic interference images was strongly developed in the eighties, starting from the case where the imaging system is exactly focused on the object surface and where both the imaging and diffusely-illuminating systems have circular apertures5 and progressing

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122 Marek J. Matczak

to the case of arbitrarily-shaped apertures,6 and further-to the case where the imaging system is arbitrarily focused with respect to the object surface.’ This gave an explanation of the influence of the diffuse illumination of the object surface on the interference-image formation in holographic interferometry,’ and it created new methods of holographic interferogram interpretation enabling one to determine the fringe-order distribution,’ the nodal loci of vibrating surfaces,l’ the 2D-displacement field,” the 3D-displacement field,5-7 the surface-strain and -rotation tensor fields,12,13 and the vectorial fields of the radius- vector and the normal of the object surface.14

The above cited methods are based on the following general equation which describes the statistically-averaged form of the light-intensity distribution Z in the interference image holographically formed in the image plane of the imaging system:

z = Zo(l + p cos 6) (1)

I0 is the light-intensity distribution in the noninterference image of the object. p is the fringe-contrast factor being, in general, the product of the Fourier transforms of the pupil function describing the shape of apertures in the imaging and illuminating systems, respectively. These transforms are functions of specific components of the displacement vector and the surface-strain and -rotation tensors. 6 is the well-known phase-difference distribution in the interference image.

Because the fringe-contrast distribution ,u contains more information about surface deformation than the phase-difference one,“” the displacement vector and the surface-strain and -rotation tensors can be directly determined from fringe-contrast measurement, without analysis of the phase-difference distribution. The important feature of HFCI is that the above mentioned tensors can be determined without differentiating displacement components.

The description of the HFCI methods does not take into account any optical noise that exists in most real measuring systems which affect the fringe visibility. The purpose of this paper is to eliminate this deficiency, to describe the practical procedure of fringe-contrast measurement, to discuss the interpretation process using an example of the determination of the displacement vector with a special HFCI method, and to evaluate its sensitivity and accuracy.

2 FRINGE-CONTRAST MEASUREMENT

Many undesirable effects decreasing the fringe visibility occur in most real measuring systems. These effects are relating to MTF of the

Holographic fringe-contrast interferometry 123

imaging system, limited temporal and spatial coherence of the light, the intensity difference between the interfering waves, the optical noise caused by grainy structure and nonlinear characteristic of holographic emulsion, and so on. All these effects can be commonly described by the fringe-contrast factor pN which modifies eqn (1):15

Z = Zo(I + /&.$ cos 6) (2)

In the case of a small aperture of both the imaging and illuminating systems, the proper fringe-contrast factor p is equal to unity and, consequently, the interference image obtained under these conditions is described by the following light-intensity distribution:

ZN = Z”(I + PN cos 6) (3)

In order to determine the distribution of the proper factor p, three light-intensity distributions, I, ZN, and &, should be known. Then, the evident relation can be used:

z - IO

The simplest way to get the distributions Z and IN is to record two adequate images to a computer memory by means of a TV camera or other image detector during hologram reconstruction. The image relating to I,, can be recorded in the same way in the case of real-time holographic interferometry. In other cases, the direct image of the object illuminated as for previous images should be recorded in the conventional way.

In practice, the distribution I0 existing in eqn (2) is in a different scale than in eqn (3). Therefore, it is necessary to make uniform the scale of Z, for all recorded images using a suitable computer program.

3 INTERPRETATION PROCESS

The interpretation process in HFCI depends, in general, on the subject of measurement and on the arrangement of the measuring system.16 In this paper, the interpretation process is exemplified for the case of the displacement-field determination. The measuring-system arrangement used for this exemplification is shown in Fig. 1. The object under investigation is illuminated by a spherical wave. The interference field relating to the change of geometrical state of the object surface is created using the double-exposure or real-time holographic technique. The interference image is formed in the image-sensor plane of the


illuminating beam

0 s

reference and reconstruction beam

IMAGING SYSTEM

Fig. 1. An arrangement of holographic measuring system for HFCI.

imaging system equipped with a rotatable narrow slit diaphragm and, in general, defocused with respect to the object surface. Then, the proper fringe-contrast factor for small deformations of the object surface takes the following form:

sin 5 PI=-

E (5)

where

5=~l{u.+(L”-L)s”[~u,+(IE+Iw)g]} (6) ”

The definitions of the quantities existing in eqn (6) are as follows:

the wavelength of the light used; the focusing distance (AF in Fig. 1) of the imaging system on the viewing direction of a current point of analysis (P in Fig. 1); the distances AP and BP, respectively (see Fig. 1); the vectorial displacement components being the normal projections of the vector of the displacement of the point P between two holographic exposures onto the planes perpendicular to the viewing and illumination directions, n, and n,, respectively (see Fig. 2); the sensitivity vector of the holographic arrangement at the point P, being the sum of the unit vectors n, and n, (see Fig. 2); the vector defining the length I and the orientation of the slit in the diaphragm used in the imaging system; the strain and rotation tensors, respectively; the operator of the oblique projection along the normal of the


Fig. 2. Normal projections II, and II, of a displacement vector u onto the planes perpendicular to illumination (n,) and viewing (n,) directions, respectively; A sensitivity

vector g as the vectorial sum of the unit vectors n, and n,.

object surface at the point P onto the plane perpendicular to the viewing direction:

n, @ n, S,=O--

wb

where 0 is the unit matrix (the identity operator), n, is the unit vector normal to the object surface, and the symbol @ denotes a tensor product which satisfies the following relation in Cartesian coordinates (4 Y, 4:

(8)

In the case where the object surface is plane, the imaging system can be exactly focused simultaneously on all points of the surface, i.e. L = L,. Equation (6) is then reduced to the following form:

Determination of the displacement component II, requires, in this case, to solve the set of two independent equations of the above type. The


following solution is then obtained:’

where cl, 1, and c2, 1, relate to the first and the second equation, respectively, 1, and l2 are perpendicular to one another, and AV is the operator of oblique projection along the normal of the diaphragm plane onto the plane perpendicular to the viewing direction:

&=I-- nA@nv

nAnv (11)

where nA is the unit vector perpendicular to the diaphragm plane. E, = +l if the fringe visibility passes through its minimum value during rotation of the slit diaphragm from its first orientation I, to the second one 12, and E[ = -1 in the opposite case.

In the case where the object has an arbitrarily-shaped surface, the set of four independent equations of the type of eqn (6) is required. Then, the solution takes the form:’

where the quantities with apostrophe have the same meaning as those in eqn (10) but they relate to the focusing distance L’ which is different from L. Ed = +l if the fringe visibility passes through its minimum value during changing of the focusing distance from L to L’ for a fixed slit orientation II, and E,_ = - 1 in the opposite case.

There are two different ways of determining the full displacement vector u. Choice of ihe way is dependent on whether the distribution of the fringe order f is known or not. In the first case u can be expressed as follows:’

(13)

where VG is the operator of the oblique projection along the viewing direction n, onto the plane perpendicular to the sensitivity vector g:

n, @ g VG=O--

“4 (14)


If the fringe-order distribution is unknown, two vectorial displacement components, II, and tii,, relating to two different viewing directions, n, and ii, are required. Then, u is given by the following expression:’

where

Ev = w {[l - (nvk>21(uv~v) + (nvfh)(nv%J(fivuv>> (16)

Of course, eqns (13) and (15) allow us to determine the full displacement vector u with indefinite sign because the double-exposed hologram does not remember which exposure was the first one and which was the second.

As it results from the above considerations, several images are required for the interpretation process. In the most complex case, twelve different images should be recorded in a computer. These images are described by the following light-intensity distributions obtained for the following conditions:

I, for the noninterference image of the object; IN for the interference image formed by the imaging system with a small aperture; Z, as above but with a slit diaphragm of horizontal orientation (II); I2 as above but of vertical orientation (I*); 1;, 1; like Z, and Z2, respectively, but by the imaging system with a different focusing distance (L’) ; &, IN,, I,, I,, fi, 1; like the above six quantities but for a different viewing direction (5,,),

Using eqn (4), where Z is successively replaced by Z1, 1,, Z; and Zi, four different distributions of the proper fringe-contrast factor are obtained: ,P~, p2, ,ui and &, respectively. Repeating this procedure for the quantities overscored with a tilde, the next four contrast distributions for a different viewing direction are obtained.

Solving eqn (5) with respect to E for values of ,LL~, p2, ,ui and & at the current point P of analysis, one obtains the values E,, g2, 5; and 5; existing in eqns (10) and (12) which allow one to determine the vectorial displacement component u, at the point P of the object surface.

The final step of the interpretation process is to determine the full displacement vector u using eqn (13) or (15).

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4 EXPERIMENTAL ILLUSTRATION

In order to illustrate the dependence of the fringe-contrast distribution on deformation of the object surface and on the imaging conditions, some experiments have been carried out in the measuring-system arrangement presented in Fig. 1. The subject of investigation was a plane plate rotated around its normal between two holographic exposures. The interference images obtained from the same double- exposed hologram but for different imaging conditions are shown in Fig. 3.

Figures 3(a)-3(d) were formed by the imaging system focused exactly on the object surface. Figures 3(a) and 3(b) have been made using a circular aperture of the small and large diameter, respectively. In the case of the small aperture, ,u = 1 and fringe visibility distribution is described only by PN, so the light-intensity distribution in this image is given by eqn (3) and it is denoted by ZN. In the case of the large circular aperture, maximum contrast occurs at the nondisplaced point where the axis of rotation crosses the object surface; here, the contrast distribution is symmetric with respect to this point. In the points where the fringe-contrast factor changes its sign, dark fringes turn into bright ones, and vice versa.

Figures 3(c), (e), (g) and 3(d), (f), (h) have been made using a narrow slit diaphragm of horizontal and vertical orientation, respectively. In this case, the proper fringe-contrast factor is described by the function (5). Because L = L, for Figs 3(c) and 3(d), the argument 5 of this function is given by eqn (9) for these images. Figures 3(e), (f) and 3(g), (h) were formed by the imaging system focused behind (L > L,) and before (L < L,) the object surface, respectively. Here the argument E is defined by eqn (6). The component being the difference between eqns (6) and (9) describes the shifting of the contrast distribution with respect to that existing for L = L,. This effect becomes visible when comparing Figs 3(d), (f) and (h).

5 SENSITIVITY AND ACCURACY OF HFCI

Determination of fringe-contrast values in HFCI is based on light- intensity measurements (see eqn (4)). Results of these measurements take discrete values because of a limited signal-to-noise ratio of an image detector and digitalization process during loading images into computer. The A/D converter should be compatible with the dynamic range of the image detector used. Thus, the number of distinguishable


intensity levels is equal to 2&, where k is the number of bits of the A/D converter. Consequently, the proper fringe-contrast factor p also takes discrete values:

PN = d--k, II =o, 1, . . . , 2k-2 (17)

It leads to discretization of &values according to eqn (5). In order to simplify the analysis of sensitivity and accuracy of HFCI,

the considerations mentioned below have been carried out for the case of the displacement-field determination concerning a plane object surface.

By virtue of eqn (9), the displacement component (denoted here by u) parallel to the slit of the diaphragm used in the imaging system can be expressed by the following equation:

(18)

The sensitivity of HFCI in this case can be defined by the smallest displacement U,in distinguished by the method. Thus, this value can be obtained by substituting 6, corresponding to p, (see eqn (17)) for c in eqn (18). Assuming that one uses the measuring system with il= 632.8 nm, L, = 300 mm and 1 = 30 mm, one obtains LL,,,~” = O-6 pm for a standard image-acquisition system with an 8-bit A/D converter, and u,,,~” = 0.08 ,um for a more sophisticated system with a 14-bit converter. It is noted that both systems are in the market.

Sensitivity of HFCI can be controlled within a wide range by changing the aperture size in the imaging system. This results from the fact that the fringe-visibility function depends on the product of the aperture size and the displacement component.

In order to evaluate the accuracy of HFCI, the following form of the relative uncertainty of the displacement component u can be used:

Au A5 AL,$l -~-+- U E L, 1

(19)

This form has been derived directly from eqn (18). The first component of the right-hand side of the above equation can be replaced by the term Aylp within the linear range of the function (5). Deriving the form of this term from eqn (4) and taking into account that AI/Z, = 21Vk, the following approximation is obtained:

AE 22-k _=-

5 II (20)

-- 10


Figure 4(a) shows the dependence of the relative uncertainty Au/u on the ratio I/I, for two different types of image-acquisition system: with &bit (curve A) and 14-bit (curve B) A/D converter, assuming L,= 3OOmm, AL, = 1 mm, 1= 39 mm, AZ = 1 pm. Figure 4(b) shows the curve B in a larger scale.

In order to compare the accuracy of HFCI with that of the methods based on the phase-difference measurement (including the phase- shifting methods), the basic equation should be discussed:

20.00

15.00

AU/U

10.00

5.00

6+U

L\ A

b- Ii

\

\

T

: 01

(21)

a

D

0.251

0:oo 0.50 1.00 1.50 2.00

I/I,

Fig. 4. (a) Relative uncertainty Au/u [%] versus the ratio I/I,, in HFCI for an image-acquisition system with 8-bit (curve A) and 1Cbit (curve B) A/D converter. (b)

Curve B in a larger scale.


where, in this case, u is the displacement component parallel to the sensitivity vector g. The relative uncertainty of u is then given by the following relation:

Au A6 Ag -=

U s+- g (22)

Because g = 2 cos (a/2), where a is the angle between the viewing and illumination directions, it can be expressed by the following approximation:

g-2[1- ($Jl’* (23)

where L, is the distance between the illumination and observation points (B and A in Figs 1 or 2). Then:

4 L L -=_ g (gL,)* l+ L, AL ( > (24)

where AL is the uncertainty of measurement of the distances L, and L,. In phase-shifting methods, being the most accurate ones, the phase difference 6 is determined from light-intensity measurements-like in the HFCI methods. So, there are no considerable differences of accuracy between the methods of HFCI and phase-shifting interferometry in applications of displacement measurements.

6 FINAL REMARKS AND CONCLUSIONS

The physical principle of HFCI consists in the analysis of speckle- pattern correlation rather than of phase difference. Therefore, no a priori knowledge (e.g. of fringe order) is required in applications of HFCI. In comparison with the phase-difference interferometry, the number of holograms required for determining 3-D displacement field can be reduced even to one. Moreover, determination of the surface- strain and -rotation tensors can be achieved without differentiating displacement field.

HFCI also allows determination of the shape of an object surface to be made. In comparison with other holographic contouring methods, this technique is considerably simpler from a technical point of view.

The sensitivity and accuracy of HFCI depend on the number of light-intensity levels distinguished by the image-detection system used. They are as good as in the best phase-shifting methods. Moreover, the


sensitivity of HFCI can be controlled within a wide range, by changing the aperture size in the imaging and illuminating systems.

HFCI can also be applied in a qualitative analysis of surface deformation. For example, the non-displaced parts or the nodal patterns can be identified as the surface areas with the best fringe contrast. The rapidity of the contrast variability during variation of the focusing distance is proportional to the magnitude of the surface strain. It enables one to discover and localize the surface regions of the minimum and maximum deformations. HFCI can also be used as an auxiliary method for identification of the zero-order fringes and determination of the fringe-order distribution.

ACKNOWLEDGEMENTS

The author wishes to express his thanks to Mr Marek Kurpiewski for his assistance in experiments. The research work leading to this paper was carried out under the Polish Central Project of Fundamental Research CPBP 02.20.

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REFERENCES

Haines, K. A. & Hildebrand, B. P. Surface-deformation measurement using the wavefront reconstruction technique. Appl. Opt., 5 (4) (1966) 595-602. Welford, W. T. Fringe visibility and localization in hologram interferometry. Opt. Commun., 1 (1969) 123-5. Walles, S. Visibility and localization of fringes in holographic interferometry of diffusely reflecting surfaces. Ark. Fys., 40 (1970) 299-403. Stetson, K. A. Fringe interpretation for hologram interferometry of rigid-body motions and homogeneous deformations. .I. Opt. Sot. Am., 64 (1974) l-10. Matczak, M. J. Fringe visibility method as a new method of holographic interferograms interpretation. In Holographic Data Non-Destructive Test- ing (D. Vukicevic (Ed)). Proc. SPZE, 370 (1982) 163-7. Matczak, M. J. Diffractional description of interference image formation and its consequences in holographic interferometry. Proc. of the European Optical Conference on Optics in Science and Technology, Rydzyna (Poland), 1983, pp. 130-139. Matczak, M. J. Single-hologram method for evaluating displacement field, In: Optical Testing and Metrology, (C. P. Grover (Ed.)), Proc. SPIE, 661 (1986) 280-5. Matczak, M. J., Pawluczyk, R. & Kraska, Z. Diffuse illumination in holographic interferometry, In: Holographic Data Non-Destructive Testing (D. Vukicevic, (Ed.)), Proc. SPZE, 370 (1982) 216-20.

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9, Kraska, Z., Pawluczyk, R. & Matczak, M. J. Identification of the local minima of fringes order in holographic interferometry. In: Holographic Data Non-Destructive Testing (D. Vukicevic, (Ed.)), Proc. SPZE, 370 (1982) 206-10.

10. Pawluczyk, R. & Kraska, Z. Diffuse illumination in holographic double- aperture interferometry, Appl. Opt., 24 (18) (1985) 3072-g.

11. Yonemura, M. Holographic measurement of in-plane deformation using fringe visibility. Optik, 63 (2) (1983) 167-77.

12. Matczak, M. J. Direct holographic determination of the rotation and strain fields by means of the fringe visibility method. In: Optical Testing and Metrology, (C. P. Grover (Ed.)), Proc. SPZE, 661 (1986) 286-9.

13. Molenda, T. M. The usable form of the left semiprojection of the displacement-gradient tensor in the fringe visibility method and its applications to evaluation of holographic interferograms. Opticu Applicutu, XVII (3) (1987) 173-8.

14. Matczak, M. J. Holographic determination of the surface shape by means of the fringe visibility method, In: Optical Testing and Metrology (C. P. Grover (Ed.)), Proc. SPZE, 661 (1986) 328-31.

15. Matczak, M. J. Szum optyczny w laserowej interferometrii kontrastu pr@k6w (Optical noise in the laser fringe-contrast interferometry). Proc. of the 2nd Symposium on Laser Technology, Szczecin (Poland), 1987, pp. 386-8 (in Polish).

16. Matczak, M. J. Fringe-contrast interferometry, In: Optics and Znformution Age (H. H. Arsenault (Ed.)), Proc. SPZE, 813 (1987) 169-70.

holographic fringe-contrast interferometry

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