1. Introduction

Despite the variety of speckle interferometric setups dedicated to the detection of mechanical deformations which has been reported in the literature since the early 70’s, sometimes with various misleading acronyms, the operating principle of speckle interferometry has been till now essentially applied by two basically different approaches.

The first one, which originate the “externally-referenced” speckle interferometers family, adopts an external reference beam [1], smooth or speckled, which is superimposed to the speckle pattern generated by the object surface and focused by the imaging system on a spatial light sensor device (at the present state of the art: a CCD or a CMOS sensor).

The external reference beam can actually be superimposed to the object beam by a beamsplitter located between the lens and the light sensor or, alternatively, between the object and the lens, as reported in Fig. 1. Alternatively, it can also be fed by a mirror (a prism or an optical fiber - renouncing to a portion of the aperture of the focusing system). In any case, when assembling an externally referenced speckle interferometer, the most important issue for obtaining a good speckle modulation is to ensure that both the imaging wavefront and the reference wavefront possess the same average curvature. The component of displacement detected by this family of speckle interferometer is exactly the same of that one given by holographic interferometry - i.e. the component along the bisector of the angle between the illumination and the observation directions.

 

Fig. 1. Speckle interferometers adopting an external reference beam realized by a beam-splitter located between: a) the lens and the light sensor; b) the object and the lens.

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The second approach permits to develop self-referenced speckle interferometers which do not require an external reference beam. The two families of speckle interferometers following this approach and which have been till now reported in the scientific literature, rely, essentially, on the “double-illumination” [2] and the “double-image” [3] principle, as depicted in Fig. 2.

The double-illumination speckle interferometers detect the component of displacement lying on the plane defined by the two illuminating directions and normal to their bisector, with no dependency from the observation direction. This family of speckle interferometers is mainly devoted to the detection of in-plane or near in-plane displacement components; for the detection of in plane components it is required that the illuminating directions form equal angles with the normal to the object surface (preferably coinciding with the observation direction). The sensitivity depends on the incident angle of the illuminating directions and it is exactly the same as it can be achieved by moiré interferometry, when adopting the same governing angles in the optical setup.

The double-image speckle interferometers actually perform an optical (and thus approximated) differentiation of the component of displacement along the bisector of the illumination and observation directions (the same component detected by holographic interferometry). The approximation (but also the sensitivity of the interferometer) depends on the amount of shear applied between the two images of the object. This family of speckle interferometers is mainly devoted to the detection of the slopes of the normal to the object surface; for the detection of pure slopes (i.e. the derivative of the out of plane displacement components) it is required that the illumination and observation directions form equal angles with respect to the normal of the object surface. By this type of interferometers it is also possible to obtain a device sensitive directly to the displacements, provided that the shear imposed between the two arms induces the interference between the investigated surface and a fixed reference surface [4]. Moreover the displacement field can be reconstructed by means of shearograms [5].

These two types of speckle interferometers, compared with the externally referenced family, do not permit in the usual operating conditions to measure directly the out-of-plane component of displacement. On the other hand they allow to assemble more simple and compact devices and, at the same time they assure, in a simple way, quite low optical path differences for the whole inspected area, thus permitting to utilize low-coherence light sources (e.g. laser diodes).

 

Fig. 2. Self-referenced interferometer based on: a) double-illumination and b) double-image principle.

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The present paper reports a completely new family of self-referenced speckle interferometers based on a quite different operating principle: the “double-focusing”. This family of speckle interferometers is capable of detecting the same displacement component given by an externally referenced speckle interferometer - as previously mentioned, the component along the bisector of the angle identified by the illumination and the observation directions, as in holographic interferometry. The operating principle of the defocusing was already used in the past in the field of interferometry for different applications [6].

With respect to a speckle interferometer fed by an external reference beam, a double-focus speckle interferometer can be implemented in a much more compact device which can be simply mounted on a video camera in place of the objective. The self-contained speckle interferometric camera thus obtained is fully portable and it can be easily interfaced with a standard laptop, through the IEEE 1394 bus, if a FireWire CCD (or CMOS) camera is employed. The interferometric camera does not need to be fed by a reference beam and it can be conveniently positioned for the imaging purposes everywhere with respect to the object under investigation, being necessary only to provide an illumination beam for the object. By adopting proper - and properly arranged - optical components, this interferometer can also be balanced, i.e. with equal optical paths, so as to make feasible the utilization of short coherence diode lasers.

Computer controlled phase-shifting techniques, widely adopted nowadays for accurate interferometric measurements, can be also easily implemented in the interferometric camera by utilizing the appropriate hardware. As it will be shown in the following, the most important requisite actually necessary for the well functioning of a double-focus speckle interferometer is that only a limited portion of the illuminated area undergoes deformations, apart from their amount - an operating condition which can be easily achieved even if whole object under inspection undergoes deformations, e.g. by placing it in front of a steady background or at the back of a windowed steady screen.

2. Operating principle of a double-focus speckle interferometer

The basic operation principle of a double-focus speckle interferometer can be easily understood by observing Fig. 3. In an imaging system working in the correctly way, as shown in Fig. 3(a), the distances which define the imaging geometry, according to the thin lens approximation, follow the well-known Gaussian equations:

where s_o and s_i are the distance between object plane and lens and between lens and image plane respectively, f is the focal length of the lens and m is the transverse magnification ratio. In this configuration, apart the aberration and the diffraction effects of the imaging system, there is a one-to-one correspondence between the points of the object plane and of the image plane. If a shorter focal length f’<f is used and the distance between the lens and the image plane is unchanged, each point of the image plane gathers the light scattered from a portion of the object plane, as shown in Fig. 3(b), or, that is the same thing, a point of the object plane spreads over an area at the image plane, as shown in Fig. 3(c). The situation is not very different if a longer focal length f’’>f is used, as shown in Fig. 3(d) and Fig. 3(e). By the layouts reported in Fig. 3 the extension of the defocused area on the object, that in the following will be referred as “reference convolving area”, can be evaluated in the approximation of the geometrical optics; the diameter of this area D ^⊗ can be easily calculated by the following formulae:

where D is the diameter of the lens. The basics of this new family of interferometer consist in using the defocused beam as the reference beam.

The speckle pattern generated by defocusing the object surface can act as a reference field provided that only a portion of the reference convolving area is interested by the deformation of the object surface; the noise introduced in the reference pattern becomes more and more severe with the growing of the extension of the deformed part of the convolving area.

In order to understand in which operating conditions a double-focus speckle interferometer can work correctly the simple specimen reported in Fig. 4(a) was considered, it is composed of a fixed part and a moving part which is tilted with the respect of the fixed one around a horizontal axis, producing, in this way, a linear variation of the out-of-plane displacements. Figures 4(b) and 4(c) reports the phase map and the correlation fringes numerically simulated which could be detected by an “ideal” speckle interferometer operating in double exposure obtained numerically by neglecting the electronic noise and the decorrelation of the speckle patterns (i.e. infinite lens aperture).

Figure 5 illustrates the degradation process of the reference pattern, which would occur when testing the specimen of Fig. 4(a) by a double-focus speckle interferometer, assuming a reference convolving area whose size is smaller than the size of the object. On the upper left part of the figure the “reference degradation factor” χ is introduced, defined as the fractional extension of the convolving area which undergoes some deformation.

At any point of the image plane the reference beam is built up by the optical integration of the speckle field over the convolving area. By splitting the integration domain A ^⊗ into the “fixed” part A_fix and the “deforming” part A_def, the reference field U_ref outcomes from the sum of two speckle fields: U_fix and U_def, whose intensities are proportional to the respective integration areas. By assuming the well known Gaussian statistics for a fully developed speckle field [7], U_fix and U_def can be represented in the complex plane, as in the lower right part of Fig. 5, whereas the standard deviations of the two Gaussian distributions are in the same ratio of the square roots of the respective average intensities (i.e. the square root of the integration areas).

In the second exposure a phase error due to the deformation of the observed surface is introduced in the reference field, that can be statistically evaluated when assuming a new uniform phase distribution of the scatterers on A_def, this gives origin to a completely new speckle field U’_def - totally uncorrelated with the previous one. It is worthwhile to notice that the assumption of a uniform phase distribution simply implies a sufficient “amount” of deformation in A_def, - i.e. practically speaking one fringe at least. The phase difference φ_e (between the new reference speckle field U’_ref, resulting from the sum of U_fix and U’_def, and the original reference field U_ref) is the phase error which is intrinsically introduced in a double-focus speckle interferometer; its standard deviation follows the well known function firstly reported in [8] and more recently in [9], which gives the relation between the standard deviation of the phase error and the amount of the speckle decorrelation. Other important contributions about the statistics of the phase error are reported in [10,11].

 

Fig. 3. Focusing and defocusing geometry: a) perfect imaging; b) light gathered at a point of the image plane for f’<f; c) defocusing of a point at the image plane for f’<f; d) light gathered at a point of the image plane for f’’>f; c) defocusing of a point at the image plane for f’’>f.

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Fig. 4. Simulation of an “ideal” speckle interferometer neglecting the electronic noise and the decorrelation of the speckle pattern: a) the deformed specimen; b) the phase map c) the correlation fringes.

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Fig. 5. Degradation process of the reference beam.

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The upper left part of Fig. 5 reports the value assumed by the parameters χ when the convolving area moves from the fixed part of the area under investigation (white zone of the lower left part of Fig. 5) to the deformed part (shaded zone). The upper right part of Fig. 5 reports the variation of the standard deviation of the phase error σ_ϕe with the parameter χ. Finally the lower right part of Fig. 5 shows in the complex plane the degradation process of the reference beam due to the defocusing process; the two Gaussian bells represent the probability density distributions of the electric field of the light scattered from the fixed part and deformed part which concur in the formation of the reference beam in the two exposures. Finally the lower left part of Fig. 5 defines the geometric entities necessary for evaluating the parameter χ.

The phase map and the correlation pattern, which could be detected in such situation represented in Fig. 6(a), are shown in the Fig. 6(b) and Fig. 6(c). In this case the degradation factor varies from point to point and assumes, inside the deforming part, the best value of 0.5 - corresponding to a standard deviation of the phase error of 1.336 - just only at the boundary between the fixed and the moving part of the specimen, and becomes 0 - corresponding to a standard deviation of the phase error of π/√3=1.814 - at a distance of one diameter form the aforementioned boundary.

If the degradation factor is constant all over the area of the specimen, the phase map and the correlation pattern reported in Figs. 6(e) and 6(f) are obtained. This can be accomplished by enlarging the convolving area till to cover the whole specimen - i.e. the diameter D ^⊗ must be made as large as the double of the maximum extension of the surface under investigation, as shown in Fig. 6(d). Finally if the specimen is positioned against a steady scattering background the illumination beam just like the object shown in Fig. 6(g), even better results can be obtained as it is shown in Fig. 6(h) and 6(i). In the quite common case of an acquisition system working in the 3:4 image aspect ratio, a constant degradation factor of 0.076 is obtained (equivalent to a phase error with standard deviation of 0.618 rad) when the object fills the whole imaged area.

Another example of fringe patterns and phase maps attainable by a double focus speckle interferometer is reported in Fig. 7, where the typical deformation field which arises on a defect like a debonding is reported. In particular Fig. 7 shows the effect of changing the diameter of the reference convolving area when the ratio between its value and the diameter of the defect assumes the values 1/2 [Fig. 7(a)], 1 [Fig. 7(b)], 2 [Fig. 7(c)] and 4 [Fig. 7(d)]. Also a multimedia file linked to Fig. 7 was generated, which shows the complete evolution of the fringe pattern and of the phase map when the diameter of the reference convolving area varies between zero and five times the debonding diameter.

The phase maps and the fringe patterns reported in Fig. 6 and Fig. 7 were obtained by adding to the speckle patterns the phase errors numerically generated from the statistical distribution which depends on the entity of deformation; in the numerical simulations the electronic noise and the decorrelation of the speckle patterns were neglected.

 

Fig. 6. Effect of the dimension of the reference convolving area: a) D ^⊗ equal to the minimum dimension of the object against a dark background; b) phase map relative to the illumination condition depicted in a); c) fringe pattern relative to the illumination condition depicted in a); d) D ^⊗ equal to the double of the maximum dimension of the object against a dark background; e) phase map relative to the illumination condition depicted in d); f) fringe pattern relative to the illumination condition depicted in d); g) D ^⊗ equal to the double of the maximum dimension of the object against a bright background; h) phase map relative to the illumination condition depicted in g); i) fringe pattern relative to the illumination condition depicted in g). The object is the small rectangle inside the bigger one in a), d) and g).

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Fig. 7. (1842 KB) Effect of the dimension of the reference convolving area for the typical deformation field of a debonding when the ratio between its diameter D ^⊗ and the diameter of the defect and is equal to: a) 1/2; b) 1; c) 2; d) 4. [Media 1]

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3. Implementation of a double-focus speckle interferometer

Several solutions are conceivable for the practical implementation of a double-focus speckle interferometer. One of the simplest and most compact solution relies on adopting a Michelson interferometer as shown in Fig. 8, in which the defocusing condition is achieved by changing the curvature of one out of two interfering beams by lens, curved mirrors or simply by unbalancing the interferometer. Figure 8(a) reports the balanced version, in which the defocusing is obtained by decreasing the focal length with a concave mirror; the same effect can be achieved by a mirror and a positive lens, while an increasing of the focal length can be obtained by a convex mirror or by a negative lens combined with a flat mirror. Figure 8(b) reports the unbalanced version, obtained by translating a flat mirror. Obviously other implementations based on the Michelson layout are possible (e.g. if the imaging optics is placed behind the beamsplitter or it is split in two stages).

By this type of interferometer it is possible to obtain a very compact implementation if all the optical components are mounted on the camera; also phase-shifting procedures can be easily applied by introducing a high-precision translator along one arm, although other solutions are available - i.e. rotating retardation laminae combined with polarizers or spatial light modulators.

 

Fig. 8. A double-focus Michelson interferometer: a) the balanced version; b) the unbalanced version. In figure: Sf, surface under investigation; ID, illumination direction; Ob, objective; BS, beamsplitter; FMr, flat mirror; CMr, curved mirror; CCD, CCD camera.

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Another simple but less compact implementation relies on the use of a Mach-Zender interferometer, as shown in Fig. 9. Also in this case the curvature of the beams can be varied by lens or curved mirrors, while the unbalanced version can be implemented by using less popular layouts. For applying the phase-shifting procedures the same remarks done for the Michelson interferometer are valid. In the solution reported in Fig. 8 the focal length of the defocused arm is decreased by a positive lens, the same effect can be attained by a substituting a flat mirror with a curved one. On the contrary the focal length can be increased by a negative lens or by a convex mirror. As for the Michelson design other implementations based on the Mach-Zender layout are possible.

The balanced versions require necessarily the use of lens or curved mirrors but, at the same time, allow to use a low-coherence light source, like the economic and compact diode lasers.

A full in-line implementation, which requires optical components designed on-purpose, can provide a very compact and simple device, as shown in Fig. 10(a). The main component of this type of implementation is the double-focus device DF which allows the interference of the focused image with the defocused one. Among the various possible solutions for this device the following ones are proposed. The first consists in using two half objectives as that represented schematically in Fig. 10(a). A second possibility is that reported in Fig. 10(b) where a smaller lens (positive in the upper row or negative in the lower row) is placed in front of the objective. The third possibility, shown in Fig. 10(c), is similar to the previous one, in this case the lens to be placed in front of the objective is designed in order to compensate the optical paths for working with a low-coherence light source. The last solution proposed, reported in Fig. 10(d) is based on the use of a particular Fresnel lens designed on purpose.

If a phase-shifting procedure must be applied in this last implementation, it is necessary to use a device able to control the phase only on a limited portion of the aperture of the overall objective, like a spatial light modulator; eventually this device can be used also as double-focus system.

 

Fig. 9. A double-focus Mach-Zender interferometer. In figure: Sf, surface under investigation; ID, illumination direction; Ob, objective; BS, beamsplitter; Ls, lens; FMr, flat mirror; CCD, CCD camera.

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Fig. 10. (a). The layout of a full in-line implementation of a double-focus interferometer. The double-focus device can be obtained by adding in front of the objective: (b) a smaller lens; (c) a smaller lens designed for compensating the optical paths; (d) a Fresnel lens. In figure: Sf, surface under investigation; ID, illumination direction; DF, double-focus device; CCD, CCD camera. In b), c) and d) the upper row refers to the use of a smaller positive lens, the lower row to the use of a smaller negative lens.

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4. Conclusions

The paper presents a family of speckle interferometer based on a new operating principle: the double focusing. This working principle, based on the extraction of the reference beam by defocusing the object wavefront, was described in details by considering the functional fundamentals and the noise introduced by the measurement technique. A simplified numerical simulation, based on the assumptions of negligible electronic noise and infinite aperture of the focusing system (i.e. no decorrelation of the speckle field), was performed in order to quantify the influence of the amount of defocusing in terms of standard deviation of the phase error. In the last part of the paper some possible implementations of a double-focus speckle interferometer were argued. Initially the possibility to use the conventional Michelson and Mach-Zender layouts was considered, and some possible solutions for obtaining the defocused reference beam were proposed. Finally a full in-line solution based on the use of particular optical elements able to provide a double focusing of the object under investigation was suggested.