## Abstract

This paper describes interference effects present in narrow band transmission acoustic microscopy images of crystals that arise from the superposition of pairs of non-aligned but symmetry equivalent Airy diffraction patterns. In the cases we examine, these Airy patterns are associated with folded portions of the slow transverse (ST) sheet of the acoustic wave surface of cubic crystals intersecting in the {110} symmetry planes. Each set of Airy fringes is inclined in the (001) observation plane to its overlapping counterpart, giving rise to curved Moiré-like fringes. Also, because of tilting of the two portions of the wave surface in opposite directions out of the (001) plane, each composite Airy fringe breaks up along its length into a large number of narrow, regularly spaced fringes, in a manner akin to wedge diffraction. We present calculations of these effects in GaAs and Ge, based on the elastodynamic Green's function, , for the infinitely extended anisotropic elastic continuum. A measured time-gated diffraction image is presented for GaAs, which is in good agreement with calculation.

## 1. Introduction

Transmission acoustic microscopy (TAM) of anisotropic solids provides considerable opportunity for studying the diffraction unfolding of wave field caustics (Every *et al*. 2004). In the far-field limit, accessed in phonon imaging experiments (Wolfe 1998), where the frequencies are typically in the region of 100 GHz, the diffraction fringes become too finely spaced to be resolved, and it is the intensity rather than the complex signal that is measured. The ray approximation, prevailing in this limit, dictates that the acoustic energy for each partial wave propagates in the direction of its group velocity ** V**, and that the intensity is inversely proportional to the product,

*γ*

_{1}

*γ*

_{2}, of the principal values of the Hessian matrix of the equation of the slowness surface, (Every

*et al*. 2004). Lines in the (

*s*

_{1},

*s*

_{2}) plane where

*γ*

_{1}

*γ*

_{2}=0 map onto caustics in the acoustic intensity, which lie in the directions of fold edges of the acoustic wave surface. Exquisite phonon focusing patterns have been reported for a number of different crystals (Wolfe 1998) revealing line and cusp caustics, and also evidence for higher order caustics (Every 1981). At the lower frequencies probed by TAM, each line caustic unfolds into a pattern of parallel fringes conforming to the Airy function

*Ai*(

*x*). Higher order caustics also have their diffraction unfoldings (Berry 1976; Poston & Stewart 1978), but that lies outside the scope of this paper.

This paper is concerned with interference phenomena present in TAM images of crystals such as GaAs, that arise from the superposition of pairs of symmetry equivalent Airy diffraction patterns that are inclined to one another. These Airy patterns are associated with folded portions of the slow transverse (ST) sheet of the acoustic wave surface which intersect in the {110} symmetry planes. Each set of Airy fringes is inclined at an angle to its counterpart in the (001) observation plane, giving rise to curved Moiré-like fringes. Also, because of tilting of the two portions of the wave surface in opposite directions out of the (001) plane, each composite Airy fringe breaks up along its length into a large number of narrow, regularly spaced fringes, in a manner akin to wedge diffraction. We present calculations of these effects in GaAs and Ge, based on the elastodynamic Green's function, , for the infinitely extended anisotropic elastic continuum, which we evaluate by the angular spectrum method using the fast Fourier transform (FFT) technique. The interference effects that we are concerned with are isolated from other overlapping diffraction structures and the contribution of longitudinal (L) waves by time gating the transmitted signal to capture only the structures of interest. In the calculations, this is done by going to high frequencies and limiting the wave vector domain over which the requisite Fourier integrals are done. There is no interference from fast transverse (FT) structures in the examples we discuss, because the relevant FT waves are shear horizontally (SH) polarized and are not coupled to (Pluta *et al*. 2003). Measured diffraction images are presented for GaAs, which are in good agreement with calculation.

Although the examples we provide are limited to GaAs and Ge, the interference effects described here are a generic phenomenon occurring in many different crystals, and not only those of cubic symmetry. The ideal Moiré and wedge diffraction effects we describe require the intersection of symmetry equivalent line caustics, but there are many other types of interference effects, which we will not even attempt to enumerate here. Moreover, these effects are not specific to acoustic waves, but can in principle occur in any wave field exhibiting overlapping Airy and other diffraction patterns.

## 2. Transmission acoustic microscopy (TAM)

In TAM (figure 1), a pair of water-coupled transducers, focused on the opposite surfaces of a sample, are used as point-like source and receiver of ultrasound transmitted through the sample (Briggs 1992; Grill *et al*. 1994; Wolfe 1998). The system operates on RF wave trains generated by a narrow bandwidth CW generator. One of the transducers is raster scanned parallel to the surface, to yield an image of the transmitted sound field. Our interest is in TAM of single crystal and other elastically anisotropic solids where, because of multiple folding of the wave surface, there are often three or more signals for each of the transverse wave branches, travelling at different group velocities in a given direction. To separate out the signals of interest, extensive use is made of time gating. The measured image of GaAs shown later in the paper in figure 5 has been obtained in this way. Images such as this reveal a wealth of diffraction phenomena (Weaver *et al*. 1993; Grill *et al*. 1994; Wolfe & Hauser 1995; Würz *et al*. 1995). For directions not too far removed from the *x*_{3} direction normal to the sample's surface, the transmitted signal corresponds fairly closely to the elastodynamic Green's function for the infinite continuum, which we discuss next.

## 3. Elastodynamic Green's function

We consider here the displacement field of a time harmonic point force of angular frequency , acting in the *x*_{3}-direction at the origin in an infinitely extensive anisotropic elastic continuum of mass density *ρ* and elastic constant tensor *c*_{ijkl}. The displacement response in the *x*_{3}-direction at a point in the observation plane, at a distance *h* from the origin, is the elastodynamic Green's function . The measured signal in TAM is, to a good approximation, proportional to , provided that focusing transducers of relatively small angular aperture are used for insonification and detection on a solid at normal incidence to its surface, since in this situation the polarization dependent angular spectrum within the solid does not differ markedly from that resulting from a time-harmonic point force acting along the *x*_{3} direction in the infinite continuum (Pluta *et al*. 2003). has the integral representation (Kim *et al*. 1994)(3.1)where the sum extends over the three acoustic branches, , * s* is the acoustic slowness vector, is the coupling factor for each mode, with

*U*

_{3}being the

*x*

_{3}-component of the polarization eigenvector, and

*V*

_{3}is the

*x*

_{3}-component of the ray or group velocity vector(3.2)where is the equation for the acoustic slowness surface, i.e. surface representing the directional dependence of the inverse phase velocity of acoustic waves in a crystal. The slowness, polarization and group velocity are determined by solving Christoffel's equation for the medium (Auld 1990).

For fixed observation plane *x*_{3}=*h* we cast (3.1) in the explicit form of a 2D Fourier transform(3.3)where *s*_{3}, *V*_{3} and *Λ*_{33} are functions of . The function , which may be considered a delayed angular spectrum pertaining to propagation from the source point to the point in the observation plane, is depicted for the slow transverse (ST) mode of GaAs(001) in figure 2*a*. Nine areas of stationary phase are visible: four maxima (at the centers of the bright circular fringes), four saddle points (centerd on the x-shaped fringes) and one minimum (the fading circular fringes at the centre).

For finite frequencies the integral (3.1) is performed numerically, using a fast Fourier transform (FFT) algorithm (Pluta *et al*. 2003). We also consider the general nature of the solution of (3.1) for frequencies tending to infinity, by invoking the stationary phase approximation.

## 4. Stationary phase approximation

In considering the far field or high frequency limit we invoke the stationary phase approximation, and confine the integration in equation (3.1) to small regions around directions in which the phase is stationary. For a given observation point , the phase is stationary when **s**_{∥} takes on a value for which the group velocity points in the direction of that point (Kim *et al*. 1994). Within the localized integration domain, the equation for the slowness surface may be approximated by in a coordinate frame with *s*_{1} and *s*_{2} oriented along the principal directions of the Hessian matrix of , and with *γ*_{1} and *γ*_{1} being the principal values of the Hessian. Assuming that *Λ*_{33} and *V*_{3} are slowly varying in comparison with the phase factor, the integral (3.1) can now be performed analytically, yielding a contribution to of the form (Kim *et al*. 1994; O'Neill & Maev 1998)(4.1)for each stationary phase point. The intensity or energy flux *I* associated with each contribution is proportional to |*G*|^{2} and thus proportional to 1/|*γ*_{1}*γ*_{2}|. The vanishing of either *γ*_{1} or *γ*_{2} causes the intensity to diverge at a caustic.

## 5. ST caustics, slowness and wave surface of GaAs

Figure 3*a* shows a grayscale intensity plot in the observation plane for the ST wave branch of GaAs, for a time gate that captures portions of the folded wave surface on either side of the 〈110〉 directions. The plot is centered on the [001] cube axis. What is observed are four pairs of symmetry equivalent line caustics, each pair intersecting at an angle of 2*α*=8.2°. Figure 3*b* shows the central region of *s*_{∥} space, with dark lines indicating where 1/|*γ*_{1}*γ*_{2}| for the ST branch diverges. The two portions enclosed in circles map onto the intersecting caustics in the top right hand quadrant of figure 3*a*. Similar regions of the *s*_{∥} plane, obtained by rotations through multiples of *π*/2 about the [001] cube axis, map onto the other pairs of intersecting caustics.

Figure 4*a*,*b* show, respectively, calculated sections of the ST sheet of the acoustic wave surface of GaAs by the (110) symmetry plane, and the nearby plane. This surface can be thought of as the locus of the end points of all group velocity vectors. Of interest here is the pair of nearby cusps in 4*b*, which are sections through two folded portions of the wave surface. As this sectioning plane is rotated towards (110), they move closer, coinciding in the symmetry plane. Figure 4*c* shows a three-dimensional global view of the ST sheet of the wave surface of GaAs, in the region of the [001] axis, giving another perspective of the intersection.

## 6. Unfolding of the line caustic into an airy diffraction pattern

Caustics are essentially a far field effect. Emerging from the far field, a line caustic unfolds into an Airy diffraction pattern, which is characterized by a set of fringes that become progressively more closely spaced with distance from the caustic (Maris 1983). To model these effects, because of the vanishing of *γ*_{1} or *γ*_{2}, requires adopting a higher order approximation to the equation of the slowness surface. Another perspective on this is provided by figure 2*b*, which shows the real part of the integrand for the ST mode of GaAs(001), plotted as a function of **s**_{∥}, where is the point of intersection of the two caustics on the [110] symmetry axis in the observation plane. Each of the two teardrop-shaped fringe patterns represents the merging of a phase maximum with a phase saddle, and thus corresponds to a higher order stationary phase point. These two higher order stationary phase points, indicated by small circles, coincide with the circled points in figure 3*b*, and map onto the point where the folds of the wave surface intersect.

Let be one of the two slowness points that maps onto . We henceforth measure position *x*_{∥} relative to . The analytical integration of (3.1) is performed around the point , regarding (*s*_{1}, *s*_{2}) as being the deviation from that point. We choose the directions of *s*_{1} and *s*_{2} to be in the principal directions of the Hessian matrix of *s*_{3}(*s*_{1}, *s*_{2}), in particular with *s*_{2} being in the direction of the vanishing principal value, which we will take to be *γ*_{2}. This direction is along the long axis of the teardrop fringe pattern in figure 2*b*. To avoid divergence of the integral (3.1), the Taylor expansion of the equation for the slowness surface is extended to cubic terms, and takes the form(6.1)In (3.1), *Λ*_{33} and *V*_{3} are slowly varying in comparison with the phase factor, and can be taken outside the integral, and so(6.2)whereThe integration with respect to *s*_{1} is performed first, and it is for that reason that all the terms depending on *s*_{1} have been collected together in the first exponential. In order of magnitude the coefficients are and , where *s*_{max} characterizes the ‘size’ of the slowness surface. At high frequencies the major contribution to the integral comes from a small region of the **s**_{∥} plane where the three terms in the curly brackets are smaller by a factor of *s*/*s*_{max} or than the preceding two terms, and so can be neglected. For the *s*_{1} integration, in completing the square one incurs a correction to the phase factor which is quadratic in *x*_{1}. This can be neglected in comparison with the phase factor already present, which is linear in *x*_{1}. The integral thus simplifies to(6.3)

Making the substitutions and , having chosen the direction of *s*_{2} so that *μ*_{03} is positive, and carrying out the integrations, one obtains(6.4)where

In a plot of the amplitude of or the intensity for a single diffraction unfolded line caustic, the phase factor has no effect, and what is observed is a diffraction pattern presenting Airy oscillations in the *x*_{2} direction, with straight continuous fringes running in the *x*_{1} direction.

Time-gated harmonic excitation and detection allows examination of specific features of . Figure 5*a*,*b* show, respectively, calculated and measured TAM time gated amplitude images obtained at a frequency of *f*=362.13 MHz on a 4.8 mm thick GaAs(001) crystal, with the detection area being 2.27×2.27 mm^{2}. The integration of (3.1) here is performed numerically with proper treatment of the transient, using a FFT algorithm, and without invoking the local approximation discussed above. The slowness, polarization and group velocity in the calculations are obtained from the solution of Christoffel's equations (Auld 1990) using the known elastic constants and density of GaAs. For higher frequencies, the numerical integration of (3.1) is performed over a limited domain around .

The measurements were done with pulse length, *t*_{p}=70 ns, detection gate, *t*_{g}=70 ns, and time delay between the detection gate and transmitting pulse, *t*_{d}=2.31 μs. In the calculations, the value of *t*_{d} was taken about 0.8 μs smaller than the experimental value, this being the transit time of the pulse through the lenses and transmission fluid. The calculated image (figure 5*a*) is in good agreement with the measured image (figure 5*b*).

In figure 5*a*,*b* there are what at first sight appear to be four equivalent sets of Airy fringes. However, on closer inspection one can make out a secondary slower variation of intensity in a direction parallel to the Airy fringes, and also the break-up of each fringe along its length into a large number of narrow, regularly spaced fringes (more obvious in figure 5*c*,*d*). Both effects are due to the interference of two Airy patterns, which are inclined to one another. These effects become more evident at higher frequencies, which can be seen in figure 6*a*,*b* which show calculated amplitude images of a (001) oriented Ge crystal, which has similar elastic constant ratios to those of GaAs. In figure 6*a* there are 4 Moiré fringes to be clearly seen, but the wedge fringes are on the whole too fine to be resolved, while in the enlarged central portion in figure 6*b* the wedge fringes are clear.

## 7. Moiré fringes

Referring back to figure 3*a*, it is evident that one needs to consider the superposition of two Airy diffraction patterns with fringes that are inclined to each other, in the case of GaAs at an angle of 2*α*=8.2. We take the *x*_{1A} and *x*_{1B} directions to be inclined at angles *α* and −*α* to the direction, and thereby parallel to the two intersecting caustics. The *x*_{2A} and *x*_{2B} directions, inclined at angles *α* and −*α* to the direction, are the directions in which the individual Airy oscillations occur. The superposition of the two non-aligned Airy functions (not considering for the moment the phase factor ) yields a signal(7.1)where

Figure 7*a* shows a grey scale representation of calculated for 2*α*=8.2. It is quite striking how the horizontal Airy fringes are suppressed by destructive interference along a set of curved Moiré-like fringes running from bottom to top. The curvature is the result of the Airy fringe spacing being non-uniform and becoming larger towards the top. The Airy function by definition is real, so the result of the superposition is real, and can be identified as the real part of the signal. Figure 7*a* resembles figure 6 in that the Moiré fringes have the same shape. Also, the Airy fringes ‘slip’ by half a spacing across a Moiré fringe, with dark fringes lining up with light. Figure 7*b* depicts the amplitude or absolute magnitude of the signal. This might have been expected to resemble more closely figure 6, which represents signal amplitude, but it looks rather different. The fringe spacing is halved, the Moiré fringes are completely dark, and the Airy fringes line up across the Moiré fringes, i.e. there is no half period slip.

In explaining the shape of the Moiré fringes, we will make use of the approximation (Abramowitz & Stegun 1965)(7.2)for the Airy function *Ai*(*z*) for *z*<−1.

With this approximation, one infers that the points of destructive interference occur where the phases of the two contributions in (7.1) differ by an odd multiple of *π*, i.e. where(7.3)Figure 8 shows the location of the Moiré dark fringes calculated from (7.3) for 2*α*=8.2. These contours coincide with the Moiré dark fringes in figure 7.

## 8. Wedge diffraction effect

The fold edges of the wave surface of GaAs are not only inclined to each other in the (001) observation plane, but are also tilted out of this plane in opposite directions through an angle (see figure 4*c*), and likewise for Ge and other such crystals. The effect of this is that along the Airy fringes, there is a rapid variation of phase in the opposite sense for the two signals being superposed, giving rise to an effect akin to wedge diffraction. One arrives at exactly the same conclusion by considering the superposition of the two Airy functions, but in this case retaining the phase factor in (6.4). The amplitude or absolute magnitude of the resulting signal is thus(8.1)where the indexes *A* and *B* relate to the circled areas in figure 2*b*. On symmetry grounds and , where *η* and *η*′ are constants, with , and so (8.1) simplifies to(8.2)

In figure 9 we show the effect of these phase shifts, or wedge diffraction effect, by plotting(8.3)taking *η*=2*π*. Figure 9 looks remarkably like figure 6*b*. Note that there are now *twice* the number of Moiré fringes there were before in figure 7*a*. The explanation is as follows, and again draws on the approximation (7.2) for the Airy function. Along the contours where the phase difference between the two sin() functions is an *even* multiple of *π*, and where in figure 7 there is a Moiré bright fringe, i.e.(8.4)the two sin() terms are identical, and on adding one obtains(8.5)

The first factor in (8.5) describes the Airy function variation, while the second factor describes the wedge-type oscillation along the Airy fringes which has a much higher spatial frequency.

Along the contours where the phases of the two sin terms differ by an *odd* multiple of *π*, and where in figure 7*a* there is a Moiré dark fringe, i.e.(8.6)the two sin() terms are of the opposite sign, and on adding, one obtains(8.7)Evidently, both (8.5) and (8.7) correspond to Moiré bright fringes, when the wedge effect is included, and both yield the same wedge-type oscillation frequency. Note that in a distance in which changes by 2*π*, the first factor, because of the modulus signs, undergoes two oscillations, as in figure 7*b*.

The Moiré dark fringes, if one can call them that, now occur where the phases of the two sin() terms differ by an odd multiple of *π*/2, i.e. where(8.8)

Along these lines the magnitude of the signal is given by(8.9)At points where either the sin() or the cos() is zero there is no wedge-type oscillation, since the dependence on *x*_{1} is or . Midway between these points, where the sin() and cos() are either equal or equal and opposite, the wedge-type oscillation has a maximum in its amplitude. There points occur where(8.10)and there are FOUR such points in an interval of 2*π*, not two. This can be seen in figure 9. Moreover, where the cos() and sin() are equal, the wedge-type oscillation varies as cos(*γx*_{1}), whereas where cos() and sin() are equal an opposite, the wedge-type oscillation varies as sin(*γx*_{1}). Inspection of figure 9 shows that this slippage of half a period in the wedge-type oscillation occurs between one Airy fringe and the next.

## 9. Distance and frequency dependence

We consider here how the different fringe spacings we have encountered above depend on *ω*, *h* and the geometry of the slowness surface.

A length characterizing the primary Airy fringe spacing is obtained by setting the argument of the Airy function in (6.4) to −1, yielding(9.1)Whilst the fringe spacing increases slowly with *h*, the angular separation of the fringes, as viewed from the source point, *x*_{2}/*h*, decreases with *h*.

A length characterizing the Moiré fringe spacing is obtained by settingand *n*=0 in (7.3). On further assuming *α*≪1, one obtains(9.2)Thus, the Moiré fringes are proportional in size to the Airy fringes, the constant of proportionality being .

The separation of the wedge diffraction fringes, obtained from (8.5) with , is(9.3)depending inversely on the frequency, and independent of the value of *h*.

## 10. Interference of non-equivalent airy diffraction patterns

The treatment above may be generalized to any case of intersection of Airy diffraction patterns, even where they are not symmetry equivalent and correspond to different group velocities. However, in such cases the contrast of the observed fringes will depend on the temporal coherence of the source. Moreover, these non-equivalent Airy patterns will in general have different amplitudes and different spacing, and this will influence the results. Thus, even if the two sets of Airy fringes happened to be parallel, there would still be a beating effect if their spacings differ. Figure 10 shows the result of calculation for the intersection of two Airy patterns inclined to each other at 20^{0} in the observation plane, with the one having spacing of 0.8 times that of the other. A wedge effect is included.

## 11. Discussion

The precision of agreement between the measured and calculated TAM diffraction patterns of GaAs reported here attests to the validity of the model and also to the accurate match between the material constants used in the calculations and those of the sample. Fitting of interference patterns of this type thus represents a sensitive tool in the determination of material constants (Würz *et al*. 1995). The model shows that while the fringes are observed at the surface, they must be present in a similar form in the acoustic field throughout the volume, and might be observable by elasto-optic scattering or other methods.

The phenomena we have described in this paper are of a general nature, and can in principle occur in any wave field for which the wave surface is folded and there are intersections between different folded portions. In the far field, the fold edges are accompanied by line caustics in the wave field intensity, which at finite frequency unfold into Airy diffraction patterns. Intersecting folds give rise to overlapping Airy patterns, which interfere with each other to produce Moiré-type fringes and something akin to an edge diffraction pattern. Evidently there is a hierarchy of structures of this sort, such as a Pearcey pattern of a cusp caustic overlapping an Airy pattern, Pearcey overlapping Pearcey, and so on. In some cases it may be appropriate to treat these as unfolded higher order catastrophes, in other cases not. Clearly there remain challenging questions to be resolved.

## Acknowledgments

F. R. N. Nabarro FRS is thanked for reading the manuscript and providing useful comments. A. G. E. acknowledges support from the National Research Foundation under Grant number 2053311.

## Footnotes

- Received December 18, 2004.
- Accepted July 4, 2005.

- © 2005 The Royal Society