Theoretical approximations for comparision with experiment

The theoretical line spread function (LSF) in the focal plane for coherent and incoherent imaging is given in the literature (Born and Wolf (1980), Gilmore et al.(1986), Kino (1987) ) . No such calculation is available for the peak detected LSF which is actually used in ultrasonic imaging. To complete the ultrasonic MTF prediction, calculations of a crude numerical nature were performed for the peak detected LSF and used to estimate the MTF of an ideal transducer. The main limitations that were imposed were those set by the software and memory (RAM) of the microcomputer used. As a result , only a basic X,Y matrix of 22 by 22 values could be used in the calculations. The results of this simulation are shown as Figure 4.8. The 3-D amplitude distribution for one quadrant of the ultrasonic beam is shown as Figure 4.8 (a). The various forms of simulated line spread function using peak, intensity and amplitude are shown as Figure 4.8 (b) . The corresponding MTFs for peak, intensity and amplitude are shown as Figure 4.8 (c). No attempt has been made to calculate the behaviour away from the focus.

Fig.4.8 Transducer response, spherically focussed (a) Calculated 3-D amplitude distribution for one quadrant

of the beam cross section.

amplitude -- intensity - peak detect 'S 0.0 CO •H a -0.2 Scan distance (mm)

(b) calculated LSF for peak, intensity and amplitude detected data. 0.8 amplitude --- intensity ... peak detect-- 0.7 •H it 0-.5 0.4

Spatial frequency (cycles/mm)

(c) Calculated MTF for peak, intensity and amplitude detected data.

143 4.4.2 Experimental results

(i) MTF at focus

The scan shown in Figure 4.6(b) was filtered using the waveforms taken along the scan. From this data, after the process of differentiation, a noisy LSF was obtained and is shown in Figure 4.9 (a). This data was smoothed only to show the possibility of beam sizing and is seen in Figure 4.9 (b) .

It should be noted that the experimental beam sizing using glass edges is considerably easier to perform than that which involves the scanning of a needle as this requires the accurate location of the needle.

The MTF for this 50 MHz transducer was evaluated using both the smoothed and unsmoothed profiles and is given as Figure 4.10. The two operations were found to give exactly the same result. This plot when compared with the crude theoretical peak detected MTF from Figure 4.8(c), shows a good resemblance. The cut-off frequency obtained matches the theoretically predicted cut-off frequency for incoherent radiation. It is seen that the experimental drop at low spatial frequencies is smaller than theoretically expected.

(ii) MTF away from the focus

Using all the assumptions given above, the experimental MTF was again filtered to extract the 52.5 MHz component. Combining the data obtained from vertical and horizontal scans, the 2-D MTF was evaluated and is given in Fig.4.11.

N O R M A L I S E D A M P L I T U D E N O R M A L I S E D A M P L I T U D E 144 1. L i n e s p r e a d f u n c t i o n f i l t e r e d f o r 5 2 . 5 MHz : NON-SMOOTHED (a) 0 T T T SMOOTHED L i n e Spr eat L F u n c t i o n (beam s i z i n g ) 1. (b) 0 200 300 40(5 100 -400 -300 -200 -100 0 distance (microns) Fig.4.9 Experimental LSF

(a) Line Spread Function for 52.5 MHz (b) Smoothed LSF, obtained from (a)

R E L A T I V E U N I T S 145 4 3 2 1 0 0 10 20 30 40 50 60 70 80

Spatial frequency (cycles/mm)

Fig.4.10 Experimental Modulation Transfer Function at focus for the transducer, filtered for 52.5 MHz

Fig.4.11 Full 3-D experimental MTF

(a) 3-D MTF for various heights around the focus

-300

0

+300

spatial frequency cycles/0.6 4m m

(b) Contour plot for 3-D MTF for various heights around focus

147

The user of this lens can therefore infer from this plot the expected degree of image degradation under the pulse-echo experimental conditions and instrument settings that are reported, when imaging impedance modulated objects, height modulated objects, or both.

The method given also provides data for the 3-D impulse response of high resolution transducers. The data can be used for 3-D image processing (Reinholdsten and Khuri-Yakub (1988))

The actual imaging capability and practical resolution limit of the lens was evaluated by scanning a 400 lines/inch electron microscope copper support grid with a line width of 15 |im. The grid was specified to have 45 |lm square holes separated from each other by 15 |im. (Agar Scientific (1991) ) . A 1mm x 1mm square scan at 10 |im step size was carried out with the grid at focus and this technique has been used by Lemons and Quate (1974). The image shown as Fig 4.12 shows that 45 p,m is at the limit of detection of this particular lens.

148

Date af scan :13/ 5/1991 Time of scan : IB:18: 9:15 Image is stored in :gr45b.Dfi' Operator/User name :ftnit Sam Sample description :ET1 Grid < Transducer frequency :50 HH* Transducer Type :U3330 Panam Pulser/RX. damping factor : Pulser/RX. energy leuel : 3 Pulser/RX. attenuation : 21

Gated Pk.Det. control : 1229 Gated Pk.Det. delay : 888 Step s u e » .010000 mm

1 . 00000mi

Fig.4.12 Acoustic image of electron microscope copper support grid. Grid has 400 lines/inch , line width of 15 |im

149 4.3 SUMMARY

A review of the possibility and the potential benefits of characterising ultrasonic transducers using the modern optical MTF concept has been presented.

The additional complexity of ultrasonic imaging has been identified and the methods required to treat these problems have been provided, leading to a 2-D MTF characteriser rather than the 1-D function used in optics. A practical procedure for the evaluation of transducer impulse response, MTF and beam sizing has been developed.

The procedure has been applied for the characterisation of a Panametrics Inc. V3330, 50 MHz Special Applications transducer for the entire focal zone of interest.

A crude approximation for the theoretical MTF of an ideal transducer with the same parameters as that considered experimentally has been presented and compared with the experimental results. An acoustic scan has been carried out on a resolution test grid that confirms the practical limit of resolution.

The characterisation obtained can be used to improve the images or to find out the best system transducer settings as seen for example in the next two points.

(i) It has been shown that peak detection is expected to yield a lower resolution image than power detection.

(ii) the need for proper matching of a transducer to the pulser-receiver has been demonstrated by two sets of MTF data, one in which zero damping had been used and another in which the transducer was better matched to the pulser- receiver. For a properly matched transducer-driving system, the MTF characterisation shows better results than the theoretically expected ones, indicating that some apodization exits under these conditions.

Chapter 5.

Correction to page 151

Paragraph 2 , line 1 should read: "In order to carry out a theoretical study, it is first important to identify the parameter that can be measured that will identify the presence of weak, b o n d s . "

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5.1 INTRODUCTION

This chapter is concerned with the theory behind this study into solid-state bonds.

In order to carry out a theoretical study, it is first important to identify the parameter that may be varied in order to identify the presence of weak bonds. The reflection and transmission coefficients are the parameters that one can measure using ultrasound and these coefficients are directly related to impedance. Measuring a change in reflection coefficient is analogous to measuring an impedance variation. Using an electrical circuit analogy, defects located anywhere throughout the bond-line thickness would act as acoustical termination sites and thus change the impedance value. A study of the change in these coefficients with impedance variation was therefore chosen to be the approach to the problem.

The compression wave or C-scanning of solid-state bonded components is in common use in industry today. The modelling of compression wave interaction with diffusion bond-lines using the simplistic model of plane-wave interaction with a perfect interface is first discussed. This model provides the reflection and transmission coefficients CR and Cx in the compression wave mode as a function of the impedance contrast between the two media.

An outline of the theory behind Rayleigh waves is provided. A Finite Difference (FD) Model has been previously developed within the Ultrasonics Group at UCL that considers the normal incidence of Rayleigh waves on welded quarter spaces once again with a perfect interface. This configuration has been studied theoretically by many workers primarily in the geophysical world when considering the propagation of pulsed Rayleigh waves across locked faults and this work shall be reviewed.

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