ASM Metals HandBook Vol. 17 - Nondestructive Evaluation and Quality Control

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Fig. 5 Schematic of a portable setup for producing a pulsed-

laser holographic interferogram of an area on a

sandwich-structure helicopter-rotor blade being subjected to fatigue stressing

Pulsed-laser systems can be subdivided on the basis of the operational mode used to generate the holographic

interferogram, that is, Q-switched and conventional pulse mode. The conventional pulsing of a flash lamp pumped solid-

state laser occurs simultaneously with the excitation of the flash lamps, so that the laser output may last for up to 2 ms. By

placing a Q-switching element in the laser cavity, short pulses (tens of nanoseconds) can be produced during a single

flash lamp cycle.

Q-Switched Pulse Mode. Double exposures using two Q-switched pulses from separate flash lamp cycles, perhaps

operationally the easiest to effect, permit the recording of an essentially static test object in two different stress states. The

distinction between this and the conventional CW holographic interferometric procedure lies in the extremely short

duration of each individual exposure, which can often obviate the need for sophisticated vibration-isolation tables. Some

stability must be retained, however, because gross movement of the object would simply result in the formation upon

reconstruction of two images (similar to a double-exposed photograph) without the characteristic fringe pattern associated

with holographic interferometric inspection. In practice, holographic exposure of the test object is recorded, a stressing

force is applied, and a second exposure is recorded, with the time interval between exposures (several seconds to many

minutes) being dependent on the mechanics of the test.

Double pulsing of the Q-switch during a single flash lamp cycle is more widely used and perhaps the best known pulsed

holographic interferometry technique. The method uses a double Q-switching action to generate both output pulses during

a single optical pumping cycle of the laser. As such, the exposure-time interval is variable from microseconds to 1 or 2

ms; the lower limit is a function of the laser characteristics and Q-switching electronics, and the upper limit is controlled

more by the characteristics of the flash lamps used to optically pump the laser. Consequently, this mode of operation

greatly reduces restrictions placed on the holographic process by virtue of the physical environment and is directly

applicable to many dynamic events. Generating two matching pulses, however, becomes more difficult as the pulse-

separation time exceeds 200 s because of the dynamic thermal conditions in the laser cavity. The result is images with

contour fringes that modulate displacement fringes, thus obscuring the information sought. In addition, the stressing force

appropriate to manifesting the flaw of interest must be capable of rapid application--a feature that may limit the

effectiveness of the method in some instances.

Conventional Pulse Mode. In a conventional pulse (non-Q-switched) mode, a long-duration (1 to 2 ms) pulse is used

for recording interferograms. This approach offers several advantages. First, the pulse duration is short enough to isolate

objects that are moving because of environmental vibrations with frequencies of 100 Hz and less, while still permitting

time-average studies of intentionally induced vibratory excitation of a component at frequencies of 1000 Hz and higher.

Second, the total energy in the laser pulse is increased by a factor of ten over a Q-switched pulse for the same flash lamp

power, permitting larger surface coverage. The most important aspect of these features is the vibration-analysis capability,

for it permits application of the pulsed technique in those situations where cyclic acoustic excitation is the preferred

method of stressing the testpiece.

Optical Holography

Revised by James W. Wagner, The Johns Hopkins University

Effects of Test Variables

In performing nondestructive inspection by optical holographic techniques, several important test variables should be

considered. Some variables, such as exposure time and reference-to-object-beam intensity ratios, are controllable by the

operator, while others are not (for example, the physical test environment). Although there are considerable

interrelationships among test variables, for the purposes of this discussion they have been divided into the following eight

categories:

• Object size

• Exposure parameters

• Surface finish

• Surface preparation

• Surface condition

• Whole-body motion

• Ambient illumination

• Environmental vibration

Object Size

There are of course theoretical limits on the size of object that can be inspected by optical holography, but generally there

are practical considerations that set the upper bounds.

Laser Coherence Length Limitations. The theoretical limitation is imposed by the finite coherence length of the

laser determined by the natural broadening of the laser spectral line. Because the instantaneous frequency of the light

leaving the laser cavity varies over some narrow range with time, light exiting the laser at one instant may not be coherent

with that emitted at some other instant. Therefore, the coherence length is the distance that light leaving the laser can

travel before there is a shift in instantaneous laser frequency sufficient to produce light that is no longer coherent with that

which left the laser initially. As a result, to produce successful holographic recordings, the path length difference between

the reference and object beams must not differ by more than that coherence length.

With essentially flat objects mounted parallel with, and close to, the holographic plate, it is necessary to consider the

difference in path length between the point on the object closest to the film plate (essentially the normal between the two)

and the point on the object farthest from the plate. Generally, if this difference exceeds the coherence length, thus

prohibiting complete holographic coverage, it is possible to move the holographic plate farther away from the object, thus

decreasing the length differential between these two points on the object. In addition, if the object itself also has

considerable depth along a path normal to the holographic plate, conventional holographic procedures will limit the

portion recorded to a distance equal to a maximum of one-half the coherence length of the laser, depending on recording

geometry.

However, because lasers with coherence lengths of the order of several meters are readily available and because

inspection by optical holography generally involves normal or near-normal observation of the surface of the object, other

practical limitations will almost always prevail in establishing maximum object size and/or surface coverage.

Vibration-Isolation Table Capacity Limitations. When a CW laser is used, the object and holographic

components must be rigidly mounted to a common fixture, which is itself isolated from the environment. This is generally

accomplished with vibration-isolation tables, which in turn place a practical upper limit on the size of the object that can

be recorded; commercial tables are available that are large enough to permit the holography of objects with several square

feet of surface area and weighing several hundred pounds. When using a pulsed laser, because vibration isolation is less

important, the holographic plate can be placed at as great a distance from the object as is desired or is physically possible.

In this situation, the practical limits of size are then set by the available energy output of the laser; commercial systems

with outputs of up to 3 to 5 J/pulse (0.8 to 1.4 mWh/pulse) are currently available, permitting coverage of areas in excess

of 9.3 m

(100 ft

) in a single hologram.

Exposure Parameters

When using a CW laser and proper vibration isolation and control over the ambient light, the exposure time is a function

of the film sensitivity, the size of the object, and its reflective properties. A proper ratio must be maintained between the

reference beam intensity and object beam intensity, which is most easily controlled with adjustable beam splitters.

(Neutral-density filters, and polarizing filters if the laser has a polarized output, can also be used in the reference beam

path for preferential control of the beam ratio.) It is generally desirable to use a reference-beam intensity somewhat higher

than the object-beam intensity, giving a ratio as measured at the holographic plate perhaps as high as 2:1 for real-time, 3:1

for double-exposure, and 4:1 for time-average techniques. (Measurement at the holographic plate automatically accounts

for the reflective properties of the object.) However, the optimum exposure time and beam ratio for a particular

component are best determined by trial and error; exposure times of seconds to minutes are not uncommon, and beam

ratios as high as 10:1 or 20:1 are sometimes used. The important consideration in judging the results of varying the

exposure time and the beam ratio is the fringe contrast in the reconstructed image, with the goal being high-contrast

fringes.

When using a pulsed laser, the exposure time is fixed by the laser characteristics. Although some control of exposure time

can be exercised when operating in the conventional pulse mode (1 to 2 ms) by controlling the laser pumping variables

such as voltage and capacitor storage, the more commonly used Q-switch systems have fixed exposure times in the 10 to

50 ns region.

Surface Finish

Investigations of surface finish have demonstrated that it imposes no severe limitations on the holographic process as

applied to the nondestructive inspection of metal parts (Ref 1). Surfaces varying from 0.1 m (4 in.) (very highly

polished) to 25 m (1000 in.) root mean square roughness (quite rough) are easily recorded with no significant effect on

interferometric fringe location. However, optimum contrast is not always obtained, particularly with polished surfaces,

which can act as specular reflectors.

Specular reflection of the light source onto the hologram reduces image visibility and causes image smearing. In

addition, specular reflections from low-roughness metal surfaces are brightest in a direction that is both a specular

direction and a direction perpendicular to the finishing direction. Either specular directions or finishing directions may

limit the area that can be recorded.

Surface Preparation

Because surface roughness imposes no severe limitations on holographic inspection, no preparation of the object surface

is required; this is one of the most important advantages of this type of inspection. However, where permissible, surface

preparation can be used to facilitate the inspection process by:

• Allowing greater area coverage and therefore reduced inspection time

• Eliminating the bothersome specular effect noted above and therefore simplifying data readout

• Producing a more uniformly bright reconstruction, which improves the o

verall fringe contrast and eases

interpretation of the reconstructed image

• Permitting greater reflectivity from nonmetallic materials such as polymers and composites

To increase the intensity of the reflected light (increasing the area coverage and often improving the fringe contrast) or

simply to minimize specular effects, any reasonably diffuse, flat, water-base white paint is generally satisfactory and easy

to remove. Still greater increases in reflected light intensity (permitting even larger area coverage or reduced exposure

time when using a CW laser) can be achieved with metallic paints or reflective paint of the type often used on highway

signs. Regardless of the paint used, it must be allowed to dry completely. If the paint structure changes between

exposures, no fringes will be produced.

Surface Condition

Because optical holographic inspection compares the surface of the test object in two states of stress, it is important that

the condition of the surface remain as constant as possible between exposures. Dust, dirt, grease, oil, corrosion, wear,

pitting, and loss of paint might be erroneously identified as displacement caused by a flaw if proper care is not exercised

during the inspection procedure.

Whole-Body Motion

Quite often, the stressing force applied to manifest a flaw as a change in the surface shape also causes a general overall

movement of the object under test. This movement is sometimes referred to as rigid-body or whole-body motion. The

movement may or may not prevent proper interpretation of the reconstructed holographic interferogram, but it can be

disconcerting. In certain situations (when performing real-time CW holography), fringe-control techniques can be used to

compensate for rigid-body motion.

Fringe-Control Mirror. One approach to fringe control requires the incorporation of an additional mirror, having two

axes of rotation, as the last element in the optical train of the object-illumination beam. If the object now rotates slightly

during application of the stressing force, a linear phase change will be introduced into the optical beam reflected to the

real-time hologram. The optical interference between this beam and the holographically reconstructed beam will

introduce a fringe pattern along the object when it is viewed through the real-time hologram. By tilting the object beam in

an appropriate manner, using the additional fringe-control mirror placed into the system, adequate compensation for the

phase change introduced by the object rotation can often be accomplished. Therefore, most of the unwanted fringes can

be eliminated, and the anomalous fringe pattern around a flaw can be better revealed (Fig. 6).

Fig. 6 Optical holographic interferograms of a clamped sandwich panel, illustrating use of fringe-

control

techn

ique to eliminate unwanted fringes. (a) Interferogram showing unwanted fringes caused by stresses from

a clamp (at top center). (b) Interferogram of the same clamped panel as in (a), but recorded with the use of a

fringe-control mirror to eliminate the clamping-

stress fringes, thus revealing a fringe pattern (at bottom center)

caused by a flaw

The appropriate setting of the fringe-control mirror can best be determined by slightly rotating the mirror about each axis

of rotation while simultaneously viewing the object through the hologram. By noting the direction of rotation that tends to

broaden and separate the fringes, the appropriate rotational direction can be found. In practice, several such adjustments

about each axis are usually required to produce the broadest spacing of fringes on the object.

Phase-Change Compensating Lens. Another method of fringe control incorporates a lens, which can be translated

along its optical axis, in the object beam path to compensate for undesirable phase changes caused by object movement

(Ref 2). Furthermore, this method can be adapted to double-exposure techniques by using a different reference beam for

each exposure and both reference beams in the reconstruction step.

Ambient Illumination

Holography is most easily performed in a darkened area. In the presence of any ambient illumination that cannot be

controlled, pulsed-laser techniques are usually the most desirable. With such techniques, the holographic plate can be

shuttered in synchronization with the laser pulse (or pulses, if used in the double-pulse mode) to hold fogging of the film

to an acceptable level.

Focal-Plane Shutters. Specially modified focal-plane shutters to provide a full opening for a few milliseconds are

quite satisfactory. This opening duration is long enough to encompass both pulses when used in the double-pulse mode

and is short enough to permit holography in almost any factory environment. Successful tests have been performed with

as high as 2.2 klx (200 ftc) of reflected ambient light, as measured at the film plane, using shutter durations of the order of

75 ms (Ref 1).

Interference Filters. If necessary, the level of ambient radiation reaching the film can be further reduced by using

optical filters matching the spectral content of the laser. Generally, a fairly broadband red filter is more than sufficient for

ruby lasers. Highly tuned interference filters, which are available to match all common laser sources, can be used to

essentially eliminate any ambient light problem. However, care must be exercised in the application of these filters

because their transmission characteristics are highly dependent on incidence angle, thus greatly restricting the permissible

optical configuration. Also, they are expensive, especially in larger sizes. However, they are useful for very specific

configurations and applications, especially in the holography of self-luminous objects.

Air Turbulence and Environmental Vibration

The holographic recording process is extremely sensitive to variations in optical path length that may occur during film

exposure. Unwanted variations in optical paths may be caused by turbulence in the air path through which beams are

propagating or by sources of rumble and vibration in the environment.

Air turbulence problems are most severe when relatively long holographic exposures are called for and can be dealt

with directly by shielding the holographic apparatus or enclosing the beam paths in a series of interconnecting tubes.

These effects can also be minimized by using optical fibers to deliver the light beams prior to beam expansion. Electronic

feedback is sometimes used to control the position of a mirror to compensate for fluctuations in refractive index caused by

air turbulence.

Environmental Vibration Problems. In most cases, air turbulence effects are less troublesome than the effects of

environmental vibration. Pulsed-laser systems have been effectively used for practical testing outside of a vibration-

controlled optical laboratory. Pulsed lasers are extremely effective in these situations and therefore allow the use of

nondestructive inspection by optical holography in most factory environments. By maintaining the pulse-separation

interval during the recording of a double-exposure holographic interferogram at or below a few hundred microseconds,

any environmental vibration effects below 1000 Hz are obviated. Although the higher-frequency vibration of objects

would hamper successful testing, such vibrations are generally of sufficiently low amplitude in normal industrial

environments to be of no consequence.

If the use of pulsed-laser systems is undesirable or if the nature of the test (such as low-frequency vibration analysis)

necessitates CW techniques, various methods are available for vibration compensation. These techniques include:

• Speckle reference beam holography (Ref 3)

• Focused-image holography (Ref 4)

• Optical processing of the reference beam (Ref 5)

• Attachment of the reference beam mirror (Ref 6) or the photographic plate itself to the object (Ref 7)

All of these techniques can generally be classed as local reference beam generation methods, in which the motion of the

object is automatically compensated for by a comparable change in the reference-beam path.

Electronic-feedback servosystem methods to provide reference beam compensation by both phase modulation (Ref 8) and

frequency modulation (Ref 9) are also available. Finally, the use of scanned-beam holography offers an approach to

compensation for object movement (Ref 10). In general, these methods require a reasonably good working knowledge of

holography because of their more complex nature and would be further troubled by any existing ambient illumination.

These object motion compensation methods, as well as fringe-control techniques, are discussed in Ref 11.

References cited in this section

1. R.K. Erf et al., "Nondestructive Holographic Techniques for Structures Inspection," AFML-TR-72-

204, Air

Force Materials Laboratory, Oct 1972 (AD-757 510)

2. L. Kersch, Advanced Concepts of Holographic Nondestructive Testing, Mater. Eval., Vol 29, 1971, p 125

3. J.P. Waters, Object Motion Compensation by Speckle Reference Beam Holography, Appl. Opt.,

Vol 11,

1972, p 630

4. L. Rosen, Focused-Image Holography, Appl. Phys. Lett., Vol 9, 1969, p 1421

5. H.J. Caulfield et al., Local Reference Beam Generation in Holography, Proc. IEEE, Vol 55, 1967, p 1758

6. V.J. Corcoran et al., Generation of a Hologram From a Moving Target, Appl. Opt., Vol 5, 1966, p 668

7. D.B. Neumann et al., Object Motion Compensation Using Reflection Holography, J. Opt. Soc. Am.,

Vol 62,

1972, p 1373

8. D.B. Neumann et al., Improvement of Recorded Holographic Fringes by Feedback Control, Appl. Opt.,

Vol

6, 1967, p 1097

9. H.W. Rose et al., Stabilization of Holographic Fringes by FM Feedback, Appl. Opt., Vol 7, 1968, p a87

10.

J.C. Palais, Scanned Beam Holography, Appl. Opt., Vol 9, 1970, p 709

11.

R.K. Erf, Ed., Holographic Nondestructive Testing, Academic Press, 1974

Optical Holography

Revised by James W. Wagner, The Johns Hopkins University

Readout Methods

When using a reconstructed interferometric image for holographic inspection, something more than a simple visual

display is generally required. A permanent record of the image, such as a photograph, is most commonly used. If further

data reduction is to be done on the interferogram, some form of electronic image is more convenient than a photographic

image. Photographic and electronic methods of readout of holographic interferograms are discussed below.

Photographic Readout

Photographic readout with a small-format camera, such as a 35 mm single lens reflex camera, is adequate for many

inspection tasks. A reflex-type camera is recommended because the picture can be composed. A wide variety of film is

available for 35 mm cameras. The film can be obtained economically in bulk form and loaded into cartridges as needed.

Slightly more expensive, but more convenient, are preloaded cartridges of 20 or 36 exposures.

Interchangeable Lenses. It is convenient, but not essential, for the camera to have interchangeable lenses. Most

needs can be met with a standard 55 mm lens--the faster the better. For photographing remote images, a telephoto lens

(for example, one with a 135 mm focal length) is often desirable. Extension tubes 25 and 50 mm lengths are an

inexpensive but useful way to obtain closer working distances from the standard 55 mm lens. Several wide-angle to

medium-telephoto macrozoom lenses are available. Generally, in making holographic reconstructions, the light levels are

fairly low, so lenses with small f-numbers are recommended for shorter exposures and smaller speckle size.

The better the quality of the photographic negative, the better the final print. For the best negatives possible, a large-

format camera, such as a 100 by 125 mm (4 by 5 in.) view camera, should be used. Such a camera also provides the

option of using Polaroid film, which is often quite convenient. A standard 135 mm lens will serve for most purposes. A

90 mm wide-angle lens and a 250 mm telephoto lens will accommodate unusual situations.

Photographic Film Specifications. The choice of photographic film is not critical. Some inadequacies in the original

reconstruction can be partially compensated for in the darkroom. For example, the recording of high-contrast, high-order

fringes in a time-average interferogram requires a dynamic range not obtainable with existing film material. To

compensate for the overbright, zero-order fringe and to enhance the weaker, higher-order fringes, a superproportional

reducer such as a 20% solution of ammonium persulfate can be used. The reducer takes silver from those portions of the

negative where there is the most, thus enhancing the weakly recorded higher-order fringes. Special high-contrast copy

films, such as Kodak 410, are available and are excellent for making reconstructions that are to be used for quantitative

data reduction. For maximum resolution in the reconstruction, moderate-speed or slow-speed films are best. In general,

the faster the film, the grainier the image. When a high f-number camera is used, however, graininess of the image will be

determined by speckle size, not film noise. For qualitative purposes or for a record only, almost any film can be used. For

convenience, high-speed films, such as Polaroid type 57 (ASA 3000), are favored.

It is not possible to record everything that can be observed in the holographic image. Lateral motion often prevents a

fringe pattern from localizing on the surface of the object. As a consequence, when a large-aperture optical system having

a very short depth of field is focused on the object, the fringes may vanish. They lie outside the depth of field of the lens

and are blurred out. To increase the depth of field, the lens aperture can be decreased, but this also increases the size of

the speckles in the resulting reconstruction.

Electronic Readout

Photographs serve well for archival purposes. However, if there is to be any further manipulation of the data, position and

density information in the photographic negative or in the image must be converted to numerical values. This can be

accomplished by using a standard television-type video system and one of several flexible, multiple-purpose systems for

electronic image processing. Analog image processors permit several types of image enhancement, including edge

enhancement, boundary detection, area determination, particle counts, and color coding of displays by density level.

The digital image processing of holographic images permits direct and automatic interpretation of fringe patterns,

resulting in psuedo-three-dimensional displays of object contour or deformation. Computer algorithms are available for

the conversion of fringes from a single interferogram image. In addition, the holographic recording and readout system

can be altered so that several images of the interferogram can be processed to produce displacement information,

providing in some cases nearly a 1000-fold improvement in sensitivity over conventional holographic interferometry.

Among these techniques are phase stepping and heterodyne holographic interferometry. The principles behind their

operation are explained in the section "Interpretation of Inspection Results" in this article.

For inspection functions, many of the standard video methods are applicable. Because the diffraction of a typical

hologram is low (<3% of the incident light), it is advisable to use a television-type video system with a camera capable of

operating at low light levels, although a standard television camera is often satisfactory. Cameras capable of operating at

low light levels are discussed in the section "Television Cameras" of the article "Radiographic Inspection" in this Volume.

The resolution available with electronic readout varies from that afforded by charge coupled device array cameras and

standard 525-line video cameras to 1000- and 2000-line high-resolution video cameras. For even higher resolution, a

special scanner-type readout system, such as an image dissector, flying spot, or laser scanner, is required. However,

because these special scanning systems generally work from photographic film inputs, their high resolution is gained at

the expense of not being able to take advantage of the real-time nature of electronic readout.

Data Storage and Retrieval. When using electronic readout, videotapes can be made for archival purposes. For rapid

response and more accurate timing, video storage disks and scan convertors can be used. In building a library of patterns,

any of the electronic storage media mentioned above can be used to advantage in conjunction with photography. For

example, a video system can be used to isolate and temporarily store a pattern of interest, which can then be photographed

from the monitor for permanent or individual storage. Photographs are much less expensive than any form of electronic

storage.

Optical Holography

Revised by James W. Wagner, The Johns Hopkins University

Interpretation of Inspection Results

The holographic process can be considered merely an optical high-fidelity recording process; the reconstructed image of

an object is indiscernible from the object seen with the unaided eye. Whatever the source, the observer sees an intensity

distribution of light that produces, using the optical system of the eye, an image of the object.

When a holographic interferogram is viewed, a system of fringes is seen overlaid on the object (Fig. 7). Different object

motions produce different fringe patterns. In general, the more complex the object motion, the more complex the fringe

pattern. The fringe pattern in a holographic interferogram of a diffusely reflecting object can be used to extract

quantitative information about the object. The procedures for interpreting the patterns depend on the type of hologram and

are presented for the most common cases later in this section.

Fig. 7 Schematic of the reconstruction process of a virtual image of an object showing fringes from a double-

exposure holographic interferogram

It is important to remember that optical holographic interferometry measures directly only surface deformation or

displacement. The information in the fringe pattern and the holographic image are directly related to the fine structure of

the surface. Should the fine structure of the surface be altered between the "before" and "after" exposures in a double-

exposure interferogram, which is the most common form of holographic interferogram, the loss of contrast in the resulting

fringe pattern will reflect that change. The general change in the shape of the test object is portrayed by the fringe pattern

itself. It is only this aspect of the interpretation of the fringes that is considered in the following discussion.

Qualitative Results

There are many cases in which quantitative results from fringe analysis are neither necessary nor desirable. For example,

qualitative information is sufficient in the inspection of laminated or sandwich structures for debonded regions. It is of no

particular significance how much the debonded region has puckered. What is important is the extent of the debonded

region and whether or not it is a debilitating flaw in the structure. The question of what size a debonded region must be in

order to qualify as a debilitating flaw is outside the realm of holography. The inspector's experience and the intelligent use

of the inspection device are what matter in this area of interpretation. The question of pattern recognition and what

constitutes a pathological or bad specimen arises. Perhaps the safest way to proceed is to compile a library of anomalous

fringe patterns from known bad or pathological specimens. Representative types of fringe anomalies include:

• Independent, localized fringe systems

• Abrupt changes in fringe continuity

• Abrupt changes in fringe density

• Significant changes in fringe shape

Depending on the amount of data or samples to be treated, the library of patterns can be further classified into types of

anomalies as experience dictates.

Out-of-Plane Displacements Versus In-Plane Displacements. In general, out-of-plane displacements lead to

fringe patterns that localize close to the surface of the object, while in-plane displacements lead to fringe patterns that do

not localize on the object. If the object and the fringe pattern are to be recorded in the same photograph, they must both be

within the depth of field of the lens of the recording camera. The concept of homologous rays is useful in understanding

fringe localization (Ref 12).

The Characteristic Function

To understand what the fringes in a holographic interferogram represent, the concept of the characteristic function is

useful (Ref 13). The distribution of the object intensity and the system of fringes observed in the interferogram can be

considered to be the product of the square of the characteristic function, M, and the image intensity distribution, I

(x), that

is:

<I(x)> = M

(x)

(Eq 1)

Wherever the characteristic function goes to zero, fringes are seen on the object. In this case, M

might well be referred to

as a fringe function. Only when the character of the object motion is well known can the characteristic function be

determined. Once the characteristic function is known, quantitative information can be extracted from the interferogram.

For step-function motion, such as that used to produce double-exposure interferograms, the characteristic function, M, is a

cosine function of position and object displacement.

Out-of-Plane Displacements

If it is known that the object of a double-exposure interferogram was in some motionless state during an initial exposure

and that it was in a different motionless state during the second exposure, then one is able directly to deduce the

characteristic function. In this situation, the characteristic function is cos (K · d/2), where K represents the sensitivity

vector and d is the vector connecting the old position, P, of a point on the surface of the object with the new position, P',

as shown in Fig. 8. The sensitivity vector is defined as k

- k

, where k

and k

are the propagation vectors of the light

scattered from the point in the directions of unit vectors n

(that is, toward the interferogram and observer) and n

(which

is on the line that emanates from the focal point of the object beam spatial filter and passes through the point on the

surface of the object), respectively. Each value of k can be determined from the formula k = (2 / )n, where is the

wavelength of the object beam.

Fig. 8 Schematic of the observation of a double-exposure holog

raphic interferogram showing the vectors and

angles relating to out-of-plane displacement of a point on the surface of the object from position P

to position

P'. See text for explanation of the symbols.

Dark fringes result whenever cos (K · d/2) = 0; that is, whenever:

(Eq 2)

Writing in scalar form and solving for the component of d parallel to K, the following expression for out-of-plane

displacements is obtained:

(Eq 3)

That is, each fringe except the first one represents a displacement of /(2 sin ) from the previous one. The first fringe

represents only half that displacement. The relations among K, k

, k

, d, , and are illustrated in Fig. 8.

The description given above is applicable in those cases when the fringes localize on the object and a definite order

number, n, is assignable to each fringe. This holds in cases of out-of-plane displacement. This is the most sensitive way to

apply holographic interferometry. Out-of-plane displacements from 15 down to 1.25 m (600 to 50 in.) can be

measured in this way. By assuming continuity and choosing the best fit for the data, displacements can be measured with

an accuracy of /25.

In-Plane Displacements

In-plane displacements can also be measured, but generally with lower precision than out-of-plane displacements. The

fringe pattern for an in-plane displacement will exhibit parallax when viewed through the interferogram. As the observer's

eye moves, the fringe pattern moves in either the same or the opposite direction with respect to a given point on the object

surface. Each dark fringe that crosses a given point during eye movement corresponds to a value of zero for the

characteristic function.

Scanning the interferogram from one side to the other, as shown in Fig. 9, defines a new vector, A = k

'' - k

'. The

displacement parallel to A can be determined from the number of fringes that are observed to cross point P. Because zero

values of the characteristic function occur at multiples of , the expression for in-plane displacements that is equivalent

to Eq 2 is: