The Foundations of Nearfield Acoustic Holography in Terms of Direct and Inverse Diffraction

David D. Bennink    Applied Measurement Systems, Inc., Bremerton, WA 98380

F.D. Groutage    Naval Surface Warfare Center, Carderock Division, Puget Sound Detachment, Bremerton, WA 98314-5215

I INTRODUCTION

The basic principles of holography were first described and demonstrated by Gabor [1,2]. In general terms, holography is an imaging method for reconstructing information concerning a three dimensional wave field from data recorded on a two dimensional surface. It follows that holography is a two-step process: a measurement or recording stage followed by a reconstruction or imaging stage. The field to be imaged is called the object wave. In the conventional approach introduced by Gabor, the measured data represents the spatial interference pattern between this object wave and a suitable reference wave. An interference pattern is recorded because it allows phase information to be included that might otherwise be lost, since detectors capable of directly measuring phase are not available for many wave fields. The actual record of the interference pattern is termed a hologram. Once a hologram is produced, it can be used to create an image through the process of direct diffraction (the solution of a direct boundary value problem for the wave equation). This is accomplished by using the hologram to modulate a suitable reconstruction wave, thus producing the necessary equivalent sources or boundary values on the hologram surface. The reconstruction wave must be appropriately related to the initial reference wave, and the image field is created on the far side of the hologram.

The nature of the boundary values on the hologram surface determines what the image field will represent in the reconstruction. Given that the hologram surface intercepted the original object wave as it propagated away from its source, two types of image field are of particular interest [3]. The first represents a continuation of the original object wave as it would have propagated without intervention. Such an image field would ideally be an exact reproduction of the object field from the location of the hologram surface onward, and could therefore be viewed and processed beyond the hologram exactly as the original object field. The second would represent the original object field from the location of the hologram surface backward to the source (excluding the volume that contained the original source since the object field would not be defined there in general). Such an image would still appear on the far side of the hologram.

The first type of reconstruction, that of determining the object field in the direction away from the source, will be called forward propagation. The second type of reconstruction, that of determining the object field in the direction toward the source, will be called backward propagation. The images for these two types of reconstruction are often referred to by other names, such as correct, true or virtual for forward propagation, and twin, conjugate or real for backward propagation. Both forms of image are produced in conventional holography. In fact, they are generated simultaneously in the reconstruction process, and may disturb one another unless separated [4,5] Both images are produced, as well as a modulated version of the reconstruction wave and other higher order terms, because the hologram is the recording of an interference pattern. For example, if the reconstruction wave was modulated directly by the complex amplitude of the object wave at the hologram surface then only the forward propagated field would be generated [6].

The images produced by conventional holography are only approximations to the correct forward and backward propagated fields. In particular, backward propagation is approximated through the process of direct diffraction. Since it is strictly necessary to correct for the effects experienced by the object wave in forward propagating from the source to the hologram surface, backward propagation is properly a problem in inverse diffraction [7]. The necessary correction can only be approximated using direct diffraction (it would strictly require the generation of an image field with sinks at unknown locations). These limitations on the conventional approach present only minor difficulties for optical holography, where the wavelengths are small compared to typical physical dimensions and the wave front propagation is essentially geometric in nature. They can cause considerably more difficulty in acoustical holography, where source size to wavelength ratios can be on the order of unity or less.

In addition, since the general idea of acoustical holography is to enable the visualization of sound fields, only the measurement step will be acoustic in nature. The reconstruction step must generate an image field that can be viewed directly. If this is done with optical processing, then increased aberrations and distortions will occur in the optical image due to the generally large difference in the wavelengths used for the two steps [6,8]. Such optical reconstructions do have the advantage of speed (real-time capability) and high capacity (large image to pixel size ratios), and many techniques have been developed for producing acoustical holograms that can be used for optical imaging [8]. All such techniques can conveniently be termed as conventional acoustic holography.

Alternate advantages, such as increased flexibility, can be gained by replacing the optical imaging methods in the conventional approach with suitable digital processing [6,9]. The data must obviously be in discrete form for such processing. If this form corresponds to direct samples of the wave field, then digital methods can be realized that go beyond the imaging techniques of conventional holography. This is possible in acoustics because both amplitude and phase information can be directly measured for a time harmonic field. Thus, it is not necessary to measure a spatial interference pattern in order to record phase information, rather the complex amplitude of the object wave can be directly recorded over the measurement surface. Such an approach eliminates the simultaneous generation of multiple images associated with hologram based reconstructions.

The flexibility gained with digital processing includes the measurement surface shape and location, which is generally restricted in conventional theory to be planar and many wavelengths from the source. For example, the theory of generalized holography describes mathematically direct diffraction imaging for arbitrary measurement surfaces, and includes a formula for backward propagation that can be implemented digitally [10]. This is again an approximation, as in conventional holography, since an exact correction for diffraction effects is not applied. However, with the digital processing of direct wave field samples it is possible to correct for diffraction in backward propagation in a more exact manner. The general approach that has developed for this in acoustics is the subject of the present work. The method will be referred to in general as nearfield acoustic holography (NAH).

Nearfield acoustic holography involves the transformation of the field between the measurement surface and a second, reconstruction surface. The transformation is based on the field satisfying the homogeneous wave equation in the volume between the two surfaces. Direct diffraction is therefore used for forward propagation, when the reconstruction surface is exterior to the measurement surface, while inverse diffraction is used for backward propagation, when the reconstruction surface is interior to the measurement surface. The first developments in this direction were naturally for field transformations between parallel planar surfaces [11,12]. For such surfaces, the processing is based on the Fourier transform, and can therefore be implemented efficiently using the fast Fourier transform (FFT) [13,14].

It has been recognized that backward propagation in the manner of NAH can provide resolution beyond the usual wavelength limitation applicable to conventional holography [15]. In principle, such enhanced resolution is possible for inverse diffraction regardless of the measurement surface location, since even farfield data is theoretically sufficient for an exact reconstruction of the acoustic field everywhere exterior to the source. In practice, enhanced resolution is obtainable with NAH because the technique can include in the processing at least some of the evanescent wave components. These evanescent wave components carry information related to the higher spatial frequencies in the field and decay in amplitude as they propagate away from the source. Therefore, in order to retain as much of this information as possible, the measurement surface in NAH is usually located close to the source, often within the extreme nearfield.

It has also been recognized that the method of NAH for planar surfaces can be appropriately extended to other suitable surface shapes, and that it can provide information concerning such derived quantities as vector intensity and power [16]. The most direct of these extensions is for cylindrical surfaces, for which the method can again be efficiently implemented using the FFT [17]. Both planar and cylindrical surfaces are, however, inherently infinite in extent. A spherical measurement surface yields the simplest application of NAH for a finite surface. Since all of these versions of NAH can be formulated as straightforward...