Double-exposure_Method

3.1.2.3 Double-exposure Method

the double-exposure method of holographic interferometry permanently keeps information about the changes of the object state. It is sometimes called the method of "frozen fringes" (Malcho, 1988).

During each exposure the light wave diffusively reflected from the object is fixed, and the deformation parameters differ from each other. The interference image characterises the change of the object wave that appeared during the period between two exposures.

By the real-time method, two wavefronts are interferometrically compared, where the first one goes out of the reconstructed hologram and the other one directly out of the investigated object that changes under the influence of deformation (Jones, Wykes, 1989). The double-exposure method is based on the interference of two reconstructed waves (Fig. 3–4). This is also the case of (as in the case of the real time method) differential interferometry where we do not recognise which of the two states is the reference one and which is the measured one.

Both holograms can be recorded on independent holographic plates where their mutual identification during the reconstruction is difficult to obtain. We try to record several holograms on one holographic plate because the information is distributed discretely rather than continuously and there is no mutual univocity of assigning any point on the hologram to any point on the recorded body.

After illumination of the hologram with the reconstructed wave, the two slightly different waves are reconstructed at the same time. As a result of their interference after travelling through the optical system we obtain a system of interference fringes on the ground glass screen. For all that, it is necessary to realise that the deformations must be so small that the geometric differences between the two images be impossible to detect visually (by the naked eye only). Both images seem to be the same but only the presence of the interference field testifies to the preceding change.

From the above relations we can obtain the expression for the illumination intensity of the reconstructed wave behind the hologram. Let us mark U₁(x, y) as the complex amplitude of the object wave and U_R(x, y) as the complex amplitude of the reference wave both creating at time t₁ in the plane of the holographic plate z = 0 the first holographic record. At time t₂ the second holographic record is exposed by means of waves U₂(x, y) and U_R(x, y). Let us mark the time difference between the two exposures Δt = t₂ – t₁. During the reconstruction, the hologram is illuminated by the reconstructed wave U_R(x, y). The complex amplitude of the reconstructed wave is proportional to the sum of the two complex amplitudes U₁(x, y) and U₂(x, y).

In the interferometric interpretation the wave U₁(x, y) represents the wave in the plane of the hologram dispersed by any object and is expressed by relation:

,(3.14)

The wave U₂(x, y) represents the wave from the moment when the object was loaded in a certain way.

,(3.15)

The small deformations of the object comparable with the laser beam wavelength influence first of all wave phase U₂(x, y) regarding wave phase U₁(x, y).

For the intensity of illumination created by the reconstructed wave behind the hologram the following must be valid:

,(3.16)

and after modification we get:

,(3.17)

This expression represents the intensity of the object illumination with the square of the amplitude, modulated by bright and dark interference fringes equivalent to the expression:

,3.18)

The bright fringes are the places where the phase difference Δφ is the even multiple of π, the dark fringes represent the places where the phase difference is the odd multiple of π. The phase change Δφ can be connected with various physical quantities such as deformation, rotation and displacement.

The double-exposure method can characterise only the field of changes concentrated around the reference state given by the hologram. The double exposure facilitates gradual recording of the changes that constantly increase in one direction so that the following reference states are successively recorded as the stages of the previous changes. Through this method a permanent record of the holographic interferogram can be made that can be further studied by other classical methods.

The advantages of the double-exposure method of recording compared to the real-time recording are (Balaš, Szabó, 1986):

• technically more easily viable,

• gives a higher contrast of the interference image,

• keeps the information about the changes in the object permanently.

The limitations of the double-exposure method of recording are:

• the changes are visible only after photochemical processing of the holographic plate and not in real time,

• it requires high stability of the optical set-up between the two exposures,

• the double-exposure hologram does not involve information about which of the two exposures corresponded to the original state of the object, which means that it is not able to determine the direction of measured displacements,

• if the aim of the experiment is to investigate various stages (for example during the successive loading) then a new double-exposure hologram must be recorded for each pair of stages while in the process of reconstruction no other combinations can be obtained,

• each change requires a new holographic plate.

The common disadvantage of both methods of recording results from the fact that differential interference techniques record all changes of the investigated objects between their immediate and initial stages. The information about them is transferred by the wave phase. In the case of displacement measurements it means for example that the final interference image not only involves the information about the displacements of the object itself but also about the mutual displacements of its individual parts, i. e. about its own deformations. It sometimes happens that changes of one type many times outnumber the other ones (its own deformations). They are the subject of investigation. We either cannot determine them at all or only with insufficient accuracy.