1. Introduction

Modern production requires effective methods for inspection and quality control at the production place. Outsourcing and globalization result in possible large distances between cooperating partners. This may cause serious problems with respect to the just-in-time exchange of information and the response to possible violations of quality standards. Consequently, new challenges arise for optical measurement techniques especially in the field of shape control, such as


• the inspection of parts where the master and the sample are not at the same place,
• the maintenance of calibration tools at different places or
• the matching of samples produced at different places.

A basic requirement for the solution of these tasks is the availability of the 3D information of the objects to be compared. In which way the object data have to be provided is determined by the measuring task itself.

In this paper we describe a new techniques for the direct comparison of the shape of two objects where it is not necessary that both samples are located at the same place. In contrast to the wellknown incoherent techniques based on inverse fringe projection this new approach prefers a coherent mask that is used for the illumination of the sample object. The coherent mask is created by Digital Holography to enable the instant access to the complete optical information of the master object at any wanted place. The transmission of the digital master holograms to this place can be done via digital telecommunication networks.

The availability of the complete optical information of the master object in the form of its digital hologram makes it possible to compare its shape and deformation with those of a sample object having a different microstructure. This can be done in two ways:


• digital comparison of the respective phase distributions of the relevant digital holograms or
• analogue comparison of both interferograms by Digital Comparative Holography.

In the case of a digital comparison the difference phases of the master object and the sample object are subtracted directly in the computer. This results in a disappearance of the object shape and the appearance of the shape difference between the two objects. In case of the analogue technique the digital interferogram of the master object has to be reconstructed by a digital light modulator such as a liquid crystal spatial light modulator or a digital mirror device. The reconstructed hologram is used for the coherent illumination of the sample object. This delivers the basis for making Comparative Digital Holography which leads to an interferogram that indicates only the difference in shape or deformation between master and sample.

2. Different Approaches for the Master-Sample-Comparison by Optical Inspection

The comparison of the shape of two nominally identical but physically different objects is a standard task in industrial inspection: the so-called master-sample-comparison. In case of objects having optically smooth surfaces, such as lenses and mirrors, holographic techniques have been applied since many years [1]. However, the optical inspection of industrial components having rough surfaces is a more complicated task. The application of interferometric techniques is limited here to rather simple objects as for instance cylindrical shells. The difficulty consists in the difference of the microstructures of the objects to be compared. Because of the strong decorrelations of the interfering wave fronts a statistical interference pattern will appear only. Therefore grazing incidence of light is preferred to reduce the interferometric sensitivity of the setup [2,3].

In recent years various methods have been developed that are based on white light fringe projection [4]. A fast comparison between master and sample can be performed if the inverse fringe pattern of the master object is used for the structured illumination of the sample object [5,6,7]. The inverse pattern can be calculated for instance by a special ray-tracing method that is based on a correspondence between the pixels of the CCD-sensor and the LCD/DMDprojector. If the sample object completely fits the shape of the master piece the inverse projection delivers an equidistant and non-distorted fringe pattern. Every deviation between master and sample causes local distortions in the fringe pattern which can be detected very fast by means of correlation and Fourier processing techniques, respectively.

The need of matching microstructures caused an important consequence for conventional holographic interferometry: the limitation to the comparison of an object with itself in two or more different states. The comparison of the shapes or the responses to a loading of two nominally identical but physically different objects made it necessary to evaluate the resulting interferograms independently and to compare the data numerically. A more elegant approach, the so-called Comparative Holographic Moiré Interferometry, was introduced by Rastogi [8] and Simova et al [9]. The method is based on the incoherent superposition of the involved interferograms and the evaluation of the resulting Moiré pattern. The appearing Moiré fringes provide a direct indication of the difference between the both objects. However, the sensitivity of this method is limited due to the poor signal-to-noise ratio in the Moiré image.

Already in 1980 D.B. Neumann published a completely new holographic technique that enables the direct detection of the deviations of two objects with different microstructure. He called it Comparative Holography [10]. The innovative aspect of this method was the coherent illumination of every state of the sample with the conjugated wave front of the corresponding state of the master. The wave front of the master plays the role of a coherent mask for the adaptive illumination of the sample. Although the procedure has interferometric sensitivity its practical relevance is still low because of the complicated experimental background. A series of valuable contributions with respect to the improvement of this technique were made by Z. Füzessy und F. Gyimesi [11] who introduced especially the application of the double reference beam technique for the independent storage and reconstruction of both object states.

In the following chapters we describe the basic principles of Comparative Holography and Digital Holography. Afterwards the combination of both methods to the new technique of Comparative Digital Holography is explained and the advantages of this new procedure are shown on example of optical shape control. Furtheron the possibility of remote shape control is discussed by making the coherent masks available at any place all over the world using the Internet.

3. Basic principles of Comparative Digital Holography

3.1 Comparative Holography
An interferometric measurement technique belongs to holographic interferometry if at least one of the object states to be compared is reconstructed holographically. In this way contours and displacements of technical objects having optically rough surfaces can be measured with interferometric precision [12]. However, the identity of the microstructures of the object surface in both states is one important boundary condition for all conventional holographic measurement techniques. Changes in microstructure causes decorrelations of the involved speckle fields with the consequence of vanishing macroscopic interference fringes. As a result of this strict boundary condition the application range of holographic interferometry is limited to the comparison of an object with itself only. But the direct indication of differences of two nominally identical objects such as their different responses due to the same load is a key problem of nondestructive testing. For instance, one of the objects can be a master without any imperfections and the other is a sample to be inspected with respect to production and material faults.

Consequently, the identification of differences between two objects requires either the separate evaluation of their interferograms and the numerical analysis of the resultant data or some tricky optical differentiation. Such an optical approach is given by Comparative Holography. For this purpose the sample is coherently illuminated in both states by the corresponding conjugated wave fronts of the master. The wave front of the master plays the role of a kind of coherent mask for the adaptive illumination of the sample. In the following the principle of the method is outlined on example of holographic displacement measurement.

In a first step a double exposure hologram is made of the master object. Between the two exposures a certain load is applied to the object. A schematic representation of a respective optical setup is shown in Fig. 1a. As it can be seen, the illumination and observation of the object is in the direction of the unit vectors e_Q1 and e_B1, respectively. The connection between the measured phase differences d(P) of the light in the object point P, its displacement d(P) and the geometry of the setup characterized by the sensitivity vector S(P) is given by the basic equation of holographic interferometry [13]:

(1)

The investigation of the sample is made in a modified optical setup where a new doubleexposure hologram is produced, Fig. 1b. Now the illumination is done from the previous observation direction e_B1 using the holographically reconstructed conjugated wave front of the master object. So the real image of the master object is imaged onto the sample. The observation of the sample is done from the former direction of illumination e_Q1. The corresponding unit vectors are:

eQ2( p )= -eB1( p )

(2)

eB2( p )= -eQ1( p )

(3)

Therefore the phase differences in the modified optical setup are:


(4)

The illumination of the respective state of the sample with the corresponding conjugated wave front of the master object displays directly the difference in deformation of both objects under the same load.

(5)

Fig. 1: Schematic representation of the paths of light during the recording of the coherent mask (a) and during the comparison with the test object (b).

3.2 Digital Holography
Digital Holography is the logical advancement of holography with respect to the availability of high resolution digital image sensors and fast signal transformations. On the other hand it permits a totally new approach for coherent optical metrology that can be described by the principle of the direct numerical wave front reconstruction. In contrast to all indirect methods that are based on the evaluation of intensity distributions digital holography allows the direct access to the relevant phase of the wave front. This opens a new degree of flexibility and practicability for holographic metrology [14,15].

Besides the direct access to the phase of the object under investigation there is another obvious advantages of the Digital Holography that follows from the digital access to the entire optical information. Consequently, digital holograms can be fed directly into digital telecommunication networks and transmitted via Internet to any place with the purpose of the 3-dimensional reconstruction of the object. This reconstruction can be done numerically in the computer or analogously by using a spatial light modulator such as an LCD or an DMD. Fig 2 shows the result of such an experiment that has been carried out by the BIAS and the Laboratory for Coherent Optics at the Humboldt University of Berlin in May 2000 [16]. The digital holograms of an experiment with a chessman subjected to thermal load have been sent to Berlin to be reconstructed with an LCD modulator.

The fast transmission of the complete 3-dimensional information of any object opens a so far unknown degree of flexibility for optical metrology. The combination of Digital Holography and Comparative Holography allows to do the master-sample comparison without the physical availability of the master at the place of testing. With the increasing globalization of economy, decreasing depth of production within the enterprises and world wide distributed production an elegant approach for shape control is imaginable that is discussed in the next chapter.

a) Picture of a chessman	b) Digital hologram	c) Digitally reconstructed intensity image
d) Digitally reconstructed interference phase of the thermally loaded chessman	e) Optically reconstructed hologram using a LCD modulator	f) Optically reconstructed interferogram using a LCD modulator
Fig 2: Digital and optical reconstruction of digital holograms that have been transmitted from Bremen to Berlin via the internet.

3.3 Comparative Digital Holography
Comparative Digital Holography uses the advantages of the Digital Holography for an effective implementation of the principle discussed in section 2.1. These advantages are:


• the direct access to the phase of the involved wave fronts,
• the inherent possibility of independent storage and reconstruction of all wave fronts,
• the possibility to compensate rigid body motions digitally with regard to a compensation of errors in the calibration between master and sample, and
• the flexible reconstruction of the separate holograms with a spatial light modulator.

The direct access to the phase of the involved wave fronts opens also a new approach to Comparative Holography in a pure digital way. In contrast to the Comparative Moiré- Interferometry [8,9] that is based on the incoherent superposition of the intensity reconstructions, this approach is based on a direct subtraction of the difference phases of both objects performed by the computer. However, a comparison of the involved intermediate states is only possible with the analogous technique. This technique uses a spatial light modulator to reconstruct the conjugated wave front of the master object. In our example we used an LCDmodulator. The technical specifications and other application possibilities of this device are discussed in [17].

The schematic setup for Comparative Digital Holography is shown in Fig. 3. The digital holograms of the master object are recorded at location A, Fig. 3a. The transmission to location B can be done via a data network (e.g. the internet). At location B the holograms are fed into a LCD-modulator. A laser reads out the holograms and reconstructs the conjugated wave front of the master object. This wave front illuminates the test object from the direction of observation during recording the master object. The observation of the test object is accomplished from the direction where the master object has been illuminated. It is essential that the separate storage and recording of all states of the objects is automatically delivered by Digital Holography. Therefore an additional reference wave for the separate coding of the respective holograms is not necessary. This property of Digital Holography reduces the technical effort of Comparative Holography considerably.

a) Recording of the coherent mask	b) Coherent illumination of the sample by the conjugated wave front of the master
Fig 3: Schematic presentation of the experimental setup for Comparative Digital Holography.

4. Experimental Verification

In this section both versions of Comparative Digital Holography are demonstrated at an example of holographic contouring. They are used for the identification of the difference between two nominally identical objects. The results will be compared with the classical holographic contouring method. The objects under test are two macroscopically identical aluminium cylinders with a cone at their upper end, Fig.4a. One of the cylinders has a small dent of some micrometers in its cone, Fig. 5b. Holographic contouring using multiple wavelengths [18] can be described by the following equation:

(6)

with L as the synthetic wavelength resulting from the wavelength l₁ and l₂ of the two holographic exposures

(7)

For the described experiment a synthetic wavelength of L= 0,218 mm was adjusted by the two single exposures with l₁=746,898 nm and l₂=749,459 nm. Dr(P) represents the relative hight deflection in point P. Because the measured phase difference d(P) is the scalar product of the sensitivity vector S(P) and the height vector Dr(P), the multiple wavelengths technique is sensitive for height differences pointing in the direction of the bisecting line of the angle between observation and illumination direction. The transformation from modulo 2p lines of equal interference phase into metric height values (e.g. the z co-ordinates of a Cartesian coordinate system) needs in general a calibration of the setup. This is required because the lines of equal phase are the intersection points of a system of rotation ellipsoids with the object. Therefore the lines of equal phase are curved and cannot be interpreted simply as contour lines.

a) Master object	b) Test object with defect in the displayed window
c) Common image of contour lines	d) Result of numerical comparison between master and test object
Fig 4: Comparative Digital Holography by numerical calculation of the difference between the interference phases of the master (a) and the test object (b).

The result of the digital comparison between the interference phases of the master and the sample is shown in Fig. 4d. The image of the sample displays only the difference of the contour between the two objects. The intensity modulation at the upper and lower site of the cone is a consequence of repositioning errors. The result of the conventional digital holographic contouring of the sample is shown in Fig. 4c. Only a careful inspection shows a small deformation of the concentric contour lines at the location of the bad spot.

The coherent illumination of the sample in both states with the corresponding real images of the master has been carried out in the setup shown in Fig. 5. The central element of the setup is the LCD-modulator. The reconstructed image of the contour lines (Fig. 6c) originates not from the numerical difference between the two interference phases but from the optical generation of the difference between the two involved states.

Fig 5: Experimental setup for Comparative Digital Holography using a LCD modulator for the reconstruction of the real images of both states of the master object and using these images as coherent masks for the illumination of the corresponding states of the test object.

The indicated difference phase d(P) corresponds to the difference of the height deflections between master and sample in the point P:


(8)

For the described experiment a synthetic wavelength of L= 0,345 mm was adjusted by the two single exposures with l₁=584,12 nm and l₂=585,11 nm.

The registered phase distribution shows in the performed experiment a high level of noise. This high level of noise is justified by the relatively large pixel size of the current available LCDmodulators compared to the available CCD-sensors (CCD 9mm, LCD 29mm). The adaptation of the space bandwidth product of the CCD-sensor and the modulator is a current matter of investigation. The manufacturers of spatial light modulators have announced the availability of LCD-modulators with a pixel size of 15mm in the near future. The comparison of picture 6b and 6c shows the advantage of the Comparative Digital Holography for contouring if the difference of two nominally identical objects with different micro structure shall be displayed in an inspection process.

a) Reconstructed intensity image of the test object	b) Mod2p-difference phase image of the test object (contour lines)
c) Image of the mod2p-difference phase as a result of the coherent illumination with the real image of the master object	d) Pseudo-3D-representation of the difference phase image including the defect
Fig 6: Comparative Digital Holography. Shape comparison between a master and a sample object by illuminating the sample object with the coherent mask of the master object. Fig a) shows the reconstructed intensity image of the test object.

5. Summary

The monitoring of the dimensional accuracy of technical objects is an important task in industrial inspection. In this article a new method has been introduced that displays directly the difference of the deformation or the contour of two objects with different microstructure by illuminating the test object with the coherent mask of the master object. The method is based on a combination of the advantages of Digital Holography with the principles of Comparative Holography. By transmitting the coherent mask with a telecommunication network Remote Shape Control can be done at any desired place without the simultaneous physical presence of the two object to be compared.

6. References

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This Paper was presented at Fringe 2001 "The 4th International Workshop on Automatic Processing of Fringe Patterns" held in Bremen, Germany, 17-19 September 2001. Proceedings edited by Wolfgang Osten, BIAS, Germany. Please contact Wolfgang Osten for full set of proceedings at wolfgang@uni-bremen.de.

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