A NEW FAMILY OF SPECKLE INTERFEROMETER
L. Bruno, A. Poggialini and O. Russo
Department of Mechanical Engineering
University of Calabria
Arcavacata di Rende (CS), ITALY 87036
ABSTRACT
The present paper describes in details the operating principle of a completely new family of speckle interferometers. These
interferometers are capable of detecting the same components of displacement given by holographic interferometry, that is the
component along the bisector of the angle identified by the illumination and the observation directions, but with no need of an
external reference beam. The working principle of this family of interferometers relies on the fact that only a portion of the
illuminated area undergoes a sensible deformation. The implementation can be indifferently carried out by adopting a
Michelson or a Mach-Zender configuration. In the paper a double focus interferometer based on the Michelson design was
used to detect the out-of-plane displacements of an artificial debond in a metallic specimen.
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 to 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 beam-splitter located between the lens
and the light sensor or, alternatively, between the object and the lens (see Fig. 1). Alternatively, it can also be fed by a mirror
(a prism or an optical fibre – 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.
a)
b)
Figure 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.
The second approach permits to develop self-referenced speckle interferometers which do not require the feeding of 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 (see 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 setup governing
angles.
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
rotations of the normal to the object surface; for the detection of pure rotations (i.e. the derivative of the out of plane
displacement components) it is required that the illumination and observation directions form equal angles with the respect of
the normal to the object surface.
These two types of speckle interferometers, compared with the externally referenced family, permit to assemble more simple
and compact devices but none of them is capable of furnishing directly the out-of-plane component of displacement. Both of
them can anyway 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. diode lasers).
a)
b)
Figure 2. Self-referenced interferometer based on a) double-illumination and b) double-image principle.
The present paper reports a completely new family of self-referenced speckle interferometers which rely on a quite different
operating principle: the “double-focus”. This family of speckle interferometers is capable of detecting the same component of
displacement of that one 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.
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 also 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 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, so 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 only requisite actually necessary for the well functioning of a double-focus speckle interferometer, is that only a limited
portion of the illuminated area undergoes a sensible deformation – an operating condition which can be easily achieved, e.g.
when the whole object under inspection stands in front of a steady background.
Operating principle of a double-focus speckle interferometer
The basic operating principle of a double-focus speckle interferometer is shown in Fig. 3, where a coaxial double focusing
element is schematically represented in combination with a light sensor device interfaced with a computer. Apart from the
practical implementation of the schematic layout of Fig. 3 into a specific device – which will be the subject of the next section –
the specific feature of the optical system of a double-focus speckle interferometer resides on the capability of forming at the
sensor plane a well focused image of the object under investigation together with a highly defocused image of the same object
– eventually including part of the surrounding background.
In Fig. 4a) an imaging system is represented where the object/lens distance so and the lens/image distance si are linked to the
focal length f – in the thin lens approximation – by the Gaussian lens equation:
1
so
+
1
si
=
1
(1)
f
where so and si can be expressed in terms of the magnification ratio m and the focal length f as:
so = f (1 + 1/m )
(2)
si = f (1 + m )
By changing the power of the lens – without changing the positions of the object, the lens and the image plane – a highly
blurred image of the surface is formed at the image plane. In particular, a generic point Po of the object is now imaged at the
front – as in Fig. 4b) – or at the rear – as in Fig. 4d) – of the image plane, where it spreads on a circle of confusion whose
diameter, for a given imaging setup, is determined by the new focal length (f’<f or f’’>f) and is proportional the lens diameter.
Figure 3. Schematic layout of a double-focus speckle interferometer.
To understand how the defocused image, when superimposed to the correctly focused image of the object, can be used as
reference beam, like in an externally-referenced speckle interferometer, it is better to consider what is actually focused at any
point of the image plane, besides the image point Pi of the corresponding object point Po of Fig. 4a). In Fig. 4c) and Fig. 4e) it
can be observed that a point located respectively at the front, Po’ , or at the rear, Po” , of the object plane is actually brought to
focus at the image plane in Pi’ ≡Pi or in Pi” ≡Pi, respectively. In both cases at points Pi’ and Pi” coherent superposition occurs
between the wavelets incoming from a circular area of the object surface (shaped by the pupil of the lens system) surrounding
the object point Po as depicted in Fig. 4c) and Fig. 4e). This area will be referred to in the following as the “reference
convolving area” – A⊗ – and it will be shown that it is the effective governing parameter for a double-focus speckle
interferometer. The diameter D⊗ of the reference convolving area can be expressed as:
D
D
⊗
⊗
⎡ f ⎛ 1 ⎞ ⎛ 1 ⎞⎤
⎜1 + ⎟ − ⎜1 + ⎟ ⎥
⎣ f' ⎝ m ⎠ ⎝ m ⎠⎦
⎡⎛ 1 ⎞ f ⎛ 1 ⎞⎤
= D ⎢⎜1 + ⎟ − ⎜1 + ⎟⎥
⎣⎝ m ⎠ f" ⎝ m ⎠⎦
= D⎢
for
f' < f
(3)
for
f" > f
where D is the diameter of the lens forming the reference speckle field. However it must be pointed out that these simple
formulas have been derived according to the thin lens approximation and by making to coincide the two focusing systems and
the two image planes. In practice, when these restraints are violated, the convolving area could be even no more centred
around the object points, without precluding the well functioning of the interferometer.
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 “moving” part of the convolving area. In Fig. 5a)
simple specimen is shown, composed of a fixed and a moving part together with the phase map and the correlation fringes,
which could be detected by an “ideal” speckle interferometer operating in double exposure; like for the intrinsic electronic
noise, always present in real acquisitions, the decorrelation process of the speckle pattern has not been taken into account –
as if the aperture diameter of the focusing system would be infinitely extended. Fig. 6 illustrates the degradation process of the
reference pattern, which could occur when testing the object of Fig. 5 by a double-focus speckle interferometer implementing a
reference convolving area whose size is smaller than the size of the object. On the left side of the figure a “reference
degradation factor” is introduced, defined as the fractional extension of the convolving area, inside which the object surface
undergoes some deformation.
At any point of the image plane the reference beam is built up by the integration of the speckle field inside the convolving area.
By splitting the integration domain A⊗ into the “fixed” part Afix and the “deforming” part Adef, the reference field Uref outcomes
from the sum of two speckle fields: Ufix and Udef with average intensities proportional to the respective integration areas. By
assuming the well known Gaussian statistics for a fully developed speckle field [4], Ufix and Udef can be represented in the
complex plane, as in the lower left part of Fig. 6 where the standard deviations of the two Gaussian distributions are in the
same ratio as the square roots of the respective average intensities (i.e. the square root of the integration areas).
a)
b)
c)
d)
e)
Figure 4. Focusing and defocusing geometry: a) Perfect imaging of a point obtained by a focal length f; b) Defocusing of a
point obtained by a focal length f’<f; c) Convolving area for f’<f; d) Defocusing of a point obtained by a focal length f”>f; e)
Convolving area for f”>f.
In the second exposure a phase error is introduced in the reference field that can be statistically evaluated when assuming a
new uniform phase distribution of the scatterers on Adef, which 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 practically
simply implies a sufficient “amount” of deformation in Adef, – practically speaking i.e. one fringe at least. The phase difference
ϕe between the new reference speckle field U’ref, resulting from the sum of Ufix and U’def, and the original reference field Uref is
the phase error which is intrinsically introduced in a double-focus speckle interferometer and its standard deviation follows the
well known function, firstly reported in [5] and more recently in [6], which gives the relation between the standard deviation of
the phase error and the amount of the speckle decorrelation.
Figure 5. Deformation geometry and corresponding simulated phase maps and intensity fringe obtained by an ideal speckle
interferometer.
Figure 6. Degradation process of a speckle pattern due to the defocusing.
The phase map and the correlation pattern, which could be detected in such situation, are shown in the upper part of Fig. 7. In
this case the degradation factor varies from point to point and assumes, inside the deforming part, the “optimal” 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. If the degradation factor is made constant all over the specimen, the phase map and the correlation
pattern represented in the central part of Fig. 7 are obtained. This can be accomplished by enlarging the convolving area till to
cover at any point the specimen the whole specimen itself – i.e. its diameter D⊗ must be made as large as the double of its
maximum extension. But if the specimen is positioned against a steady background, scattering the illumination beam just like
the object, even better results can be obtained as it is shown in the lower part of Fig. 7, where a degradation factor of 0.064 is
obtained giving a standard deviation of the measured phases of 0.576. 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) when the object fills the whole imaged area.
Implementation of a double-focus speckle interferometer
Quite different solutions are conceivable for implementing a double-focus speckle interferometer. A “full in-line” solution can
originate the most simple and compact devices; a few possible full in-line implementations are depicted in Fig. 8. For instance,
a secondary focusing system can be obtained by adding a smaller positive, or negative, lens at the front of the imaging optics
as in Fig. 8a), thus increasing, or decreasing, the focal length of the primary focusing system. The equalization of the optical
paths can be also optimized by using an on-purpose fabricated optical component as the one shown in Fig. 8b). Even more
sophisticated solutions can be conceived, based on a modified Fresnel lens as shown in Fig. 8c) or diffractive optical
elements, which could deliver a reference speckle pattern with the same grain size and effectively shaped as the object
speckle pattern. The introduction in the optical path of a programmable spatial light modulator, at the front or at the rear of the
focusing optics, could assure a much higher versatility to the interferometer but, at least at the present, with much higher costs.
On the counterpart only the use of a spatial light modulator could ensure the implementation in the optical system of a phaseshifting technique in a full in-line double-focus speckle interferometer.
Figure 7. Phase map and the correlation pattern for different level of degradation factor.
a)
b)
c)
Figure 8. Full on-line double focus systems obtained by placing at the front of the objective: a) a smaller lens; b) a smaller lens
mounted on a plate to equalize the optical paths; c) a modified Fresnel lens.
On the hand the operating principle of this new type of speckle interferometer can be effectively implemented by using
standard optical components when adopting a Michelson or a Mach-Zender configuration. Figure 9a) shows a very simple
solution, where a concave mirror is the sole optical elements introduced in place of the flat mirror in a standard Michelson
interferometer. Without doubt, the Michelson configuration assures the design of more compact devices by using concave
mirror for decreasing the focal length along one arm of the interferometer, or alternatively a convex mirror for increasing the
focal length. A converging or diverging lens placed in front of the plane mirror can be respectively employed for obtaining the
same results by using more easily available optical items as shown in Fig. 9b). In any case the aberrations introduced in the
secondary focusing system do not imply degradation in the performances of the interferometer. The calculus of the effective
size of the reference convolving area – i.e. the plane focused by the secondary focusing system – must be easily carried out
by the use of an optical design software as shown in Fig. 10. During the design process, the interferometer can also be
balanced as accurately as possible for making feasible the use of short coherence light sources. Unfortunately it is impossible
to manage the ratio between the average intensities of the reference and object speckle field, without including more
components in the setup, e.g. properly arranged polarizers included in the optical paths of the interferometer. As it will shown
in the next paragraph this fact does not actually constitute a problem for the well functioning of the interferometer, even when
acquiring simply a fringe correlation pattern by light intensity subtraction. Phase-shifting techniques can also be implemented
by any of the usual and well-assessed solutions (e.g. PZT actuators), which are nowadays widely adopted in the field of optical
metrology.
Figure 11a) shows a compact device implementing a double-focus speckle interferometer based on the layout of Fig. 10,
where the concave mirror has been replaced by a plano-convex lens positioned against a flat mirror. The interferometer has
been assembled on a 40x40x40 mm cube Linos model 06-1081 directly mounted on a Sony Firewire CCD camera model
XCD-X710 with a ¾ aspect ratio 1024x768 sensor and a pixel size 4.65x4.65 µm. The focusing lens is an achromatic doublet
Linos f60 model 32-2323 (clear aperture Φ21 mm) mounted on a focusing system. The beamsplitter is a non-polarizing
broadband 20x20x20 cube. For alignment purposes the flat mirror is mounted on a 3 degrees of freedom tilting platform Lions
model 06-5089. The extension of the reference convolving area, expressed as the ratio between its radius and the diagonal of
the imaged area, assumes values about the unity when the magnification ratio varies between 1/100 and 1/10, corresponding
to an imaged area varying from 480x360 to 48x36 mm2. For the sake of completeness an unbalanced Michelson
interferometer are reported in Fig. 11b), mounting the same objective and a CCD camera with the same sensor size but with a
lower resolution. By this approach the reference convolving area can be simply controlled by moving the mirror of the
secondary focusing system back or forth from the equal path configuration; in this operating conditions the average intensities
of the interfering speckle fields could become too different to provide good fringe quality.
a)
b)
Figure 9. Implementation of double focus interferometer based on the Michelson design. The defocusing is obtained by: a) a
concave mirror; b) a plano-convex lens placed in front of the mirror.
Figure 10. Ray tracing of a balanced double focus interferometer based on the Michelson configuration of Fig. 9a).
a)
b)
c)
Figure 11. Double focusing interferometers based on the Michelson design in the: a) balanced configuration without
illumination; b) balanced configuration with a laser diode illumination system; c) unbalanced configuration.
The balanced and unbalanced Michelson interferometers were used to retrieve the out-of-plane displacements of a debonding
(the typical defect found on a composite laminate or on a painting) artificially introduced in the specimen reported in Fig. 12.
The defect diameter, corresponding to the diameter of the pressurized chamber is 1”, while the size of the whole specimen is
5”x5”.
The fringe patterns obtained by the two types of interferometer are reported in Fig. 13 and Fig. 14. In particular Fig. 13 shows
the fringe patterns obtained by the balanced interferometer working at the magnification ratio of about 1/10 when using a NdYAG laser (Fig. 13a)) and a diode laser (Fig. 13b)).The entity of the pressure was varied in order to obtain the same number of
fringes for both lasers. Figure 14 reports the fringe patterns obtained by the unbalanced interferometer working at the
magnification ratios of about 1/10 when increasing the extension of the convolution area. It must be noticed in both cases the
good quality of the fringe pattern even when the size of the defect becomes comparable to the imaged zone of the test object.
Figure 12. The specimen realized for simulating the displacement field of a defect like a debonding.
a)
b)
Figure 13. Fringe patterns obtained by the balanced interferometer working at a magnification ratio of about 1/10 when using:
a) a Nd-YAG laser; b) a diode laser.
a)
b)
c)
Figure 14. Fringe patterns obtained by the unbalanced interferometer working at a magnification ratio of about 1/10 obtained
by increasing the extension of the convolution area.
Conclusions
The paper presents a new family of interferometers working by a completely new operating principle: the double focusing. A
comprehensive description of various interferometer configurations are also reported. This type of interferometer can be
implemented either in the Michelson or Mach-Zender configuration, in both cases it is possible to realize a balanced or
unbalanced version of the interferometer: in the former case a low-coherence laser can be employed, in the latter the practical
implementation is easier. The well functioning of a double focusing speckle interferometer is achieved only when a limited
portion of the illuminated area undergoes a sensible deformation.
The use of this type of interferometry is mainly addressed to the field non-destructive testing, being able to provide more
straightforwardly interpretable experimental results than the classical shearography. Hence a double focus device can be
advantageously applied to detect debonding between skin and honeycomb of a composite laminate, propagation of flaws in
mechanical components, defects in artworks and architectural buildings. For these investigations the presence of the
aforementioned entities can be emphasized by classical loading systems, such as acoustic or thermal methods. Finally it must
be pointed out that this type of interferometer can be used for the detection of the nodal lines when analysing vibration
phenomena by time-averaged techniques.
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