Resolution of scanning ultrasonic imaging systems with arbitrary transducer excitation

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Resolution of scanning ultrasonic imaging systems with arbitrary transducer excitation

M. Nikoonahad, E.A. Ash

To cite this version:

M. Nikoonahad, E.A. Ash. Resolution of scanning ultrasonic imaging systems with arbitrary trans-

ducer excitation. Revue de Physique Appliquée, Société française de physique / EDP, 1985, 20 (6),

pp.383-389. �10.1051/rphysap:01985002006038300�. �jpa-00245348�

Resolution of scanning ultrasonic imaging systems ^with arbitrary

transducer excitation

M. Nikoonahad and E. A. Ash

Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WC1E 7JE, U.K.

(Reçu ^{le 7} septembre ^1984, révisé le 6 mars 1985, accepté ^{le 12 mars} 1985)

Résumé.

On propose ^une formulation analytique permettant ^de prévoir ^les performances ^de systèmes imageurs

ultrasonores large ^bande.

L’effet de la dépendance ^en fréquence de l’atténuation sur la résolution de l’image ^est analysé. ^Des exemples

concernant le cas d’une impulsion radiofréquence ^{étroite et d’une} excitation large ^bande balayée linéairement

en fréquence ^sont présentés.

On montre que dans la mesure où seule la résolution de l’image ^est considérée, ^{il est} possible ^{de faire une très}

bonne estimation intuitive de la « fréquence ^continue équivalente ^».

Abstract.

An analytical formulation is presented ^which predicts ^the imaging performance ^{of broadband ultra-}

sonic imaging systems. The effect of frequency dependent attenuation on image resolution is analysed. Examples

of the case of a narrow RF pulse ^{and a} wideband FM chirp pulse excitation are presented. It is shown that as

far as the imaging resolution is concerned it is possible ^{to make a very} good ^{intuitive estimate of the «} equivalent

cw » frequency.

Classification

Physics ^Abstracts

43.35

1. Introduction.

The resolving power of imaging systems ^{as a function} of the wavelength, À, ^was established by ^{Abbé over a} century ago. The modification of the result for _spa-

tially ^cohérent, incoherent or partially coherent radia- tion is also well known [1]. The extension to the ^case when the incoherent illumination ^{covers a} broadband

of wavelengths

^{as for} example ^{the case of an} optical microscope illuminated by ^a tungsten ^{source is} quite simple. ^{One can} then define a separate point spread

function for each, ^narrow band of radiation. The resultant point spread function is obtained by ^the

addition of the corresponding intensities.

In acoustic imaging ^{one can however encounter a}

situation where the illuminating signal ^is broadband,

but where there is still complete temporal coherence,

in the sense that the waveform is fully defined, ^{as is the} corresponding complex frequency spectrum (1). ^In

this situation the intensity point spread function is obtained by ^the complex addition of the amplitude

(1) ^{It is} possible ^that with the emergence of femto- second optical pulses, ^{the same} considerations might ^be

needed in analysing optical focussing systems.

point spread functions for each component ^{in the}

frequency spectrum. ^In medical ultrasonics and in systems for nondestructive examination it is not uncommon to encounter bandwidths which exceeds

one octave. Indeed, ^{since in} many situations the aim is to achieve the greatest possible range resolution,

there is always ^{an incentive to use shorter} pulses ^and

hence wider bandwidths. In acoustic microscopy [2-4]

there is also a tendency ^{to work with ever shorter} pulses, ^or alternatively [5, 6] ^with broadband, ^coded pulses. ^It is in such situations that a more rigorous

calculation of the effective resolution is needed, ⁱⁿ

order to make rational choices in instrument design.

The issue is of particular importance in acoustic

imaging systems, where the attenuation of the signal

increases with frequency

often with the _square of

frequency. ^{In thermal wave} imaging systems [7, 8]

where the real and imaginary part ^{of the} propagation

constant are equal, ^this problem ^is even more severe.

In such situations the judicious ^{choice of an illumina-}

tion frequency spectrum designed ^to optimise ^the performance ^{of the} imaging system, ^is by ^{no means}

obvious.

In this _paper we will develop ^the theory ^needed

to calculate the effective resolution of an acoustic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01985002006038300

384

imaging system

^{i.e. the effective} point spread ^func-

tion

and then show how this affects the imaging performance ^{of a} system ^for short R.F. pulses ^{and for}

linear FM chirp pulses. ^{We will show that the results}

are in accord with a simple ^intuitive concept ^{of an} equivalent monochromatic illumination at a suitably averaged frequency.

2. Theory.

A scanning imaging system ^is illustrated in figure ^1.

The aperture ^Tx produces ^{a field onto the} object

and the aperture ^Rx gathers the scattered and trans- mitted radiation. The parameters ^{of the two} aper- tures can in general ^be différent, though, they ^{are of}

course identical in a reflection version, ^{where one} aperture only ^is used. The analysis ^{of two} systems, however, proceed ^{in the} same manner. We assume

that a complex ^scalar phasor adequately ^describes

the two-dimensional field distribution between the transmitter and receiver pupils.

Fig. ^1.

^The general ^{form of a} transmission scanning imaging system.

Our aim is to derive the amplitude ^{of the} output signal ^from the instrument U (xs; ys, f ) ^{when the} system ^is illuminated at a frequency f. ^Here (xs, ys)

are the object ^scan coordinates. The output ^will depend ^{on the} point spread functions, ^{at the} object plane, ^{due to} transmitting ^{and to} receiving pupils, T(xo, yo, f ) ^and R(xo, yo, f ) ^{and on the} object

function 0(xo, yo). We will for the moment assume

that the object function is independent ^of frequency.

The function U(xs, ys, f ) ^{is obtained} using ^{the well-}

known convolution [9, 10], expression,

The transmitter point-spread ^{function can be} regarded

as based on the geometrical ^functions Tg(xo, ^{yo, f),}

which one would obtain in a lossless medium, ^when

the lenses are excited at a frequency f ^{of unit} ampli-

tude. It also depends ^{on the} complex excitation _spec- trum E(f), ^{and the} frequency-dependent attenuation of the medium, Wt(f). ^In general the diffraction function Tg(xo, ^{yo, f )} and attenuation function Wt(f)

can not be completely separated from each other.

However, ^{we have} previously ^shown [11] ^{that the}

effect of a finite attenuation on the field distribution

(and ⁱⁿ particular ^{on focal} plane distribution of an

acoustic lens) ^at frequencies of interest is _very small.

We therefore have :

The corresponding ^receiver point spread function, R(xo, yo, f ), ^is given by :

where Rg(xo, ^{yo, f )} is the receiver geometrical point spread function and Wr(f) is the fluid weighting

function for the propagation ^{from the} object ^to the

receiver.

For a thin lens, Tg(xo, ^{yo, f ) and} Rg(xo, ^{yo, f)}

take the form of the two-dimensional Fourier trans- form of the pupil ^functions [9, 10]. ^The precise ^nature

of the Wt(f) ^and Wr(f) ^{depend on the model} [12]

one chooses to represent absorption mechanism in the fluid. Irrespective ^{of the} origin ^{of such} frequency- dependent ^losses, there exists an associated dispersion

characteristic which _may be found by ^the application

of the Kramer-Kroning relationship.

Equation (1) ^is recognized ^{as a two} dimensional convolution of the product of transmitter and receiver

point spread functions and the object ^{function. For}

a wideband system ^{we obtain a set of} images, ^each

of which is the result of a convolution operation ^at

a given frequency. ^In view of the linearity ^{in the}

system, ^the broad-band, ^scanned image amplitude, U(xS, Ys) ^{is the} superposition of all the images ^{in the} frequency ^{band, so that}

where the excitation spectrum is contained within the limits fi ^to f2. Changing ^{the order} of integration, ^{we obtain}

so that

Equation (6) ^{is a} convolution integral ^{which we can write more} compactly ^as,

where

We recognize He(x0, yo) ^as the effective point spread

function of the complete system. This broadband

point spread function will lead to a certain image.

It is at once clear that the resolution in this image ^will

be better than that which would be associated with a

monochromatic system ^{excited at} the lowest frequency

in the spectrum ^{and worse} than that associated with the

highest frequency. ^{For the case of a lossless} coupling fluid, ^one might expect ^the single excitation frequency which, ^{from the} standpoint ^of resolution, gives ^a per- formance equivalent ^to E(f), ^{to be some} weighted

mean of all the frequencies present. ^{It seems intui-}

tively plausible ^{that the effective} frequency will ^be

somewhere near the centre of gravity of the energy spectrum. Coupling losses in the liquid, ^{which rise} rapidly ^with frequency ^will de-emphasize ^the higher frequencies ^{in the} spectrum, the effective spectrum in the _presence of losses is,

The centre of gravity ^{of the} spectrum fc ^{is then} given by :

In the following ^we will find it useful to compare the actual point spread function, ^as computed ^from equation (8), with that arising ^{from as assumed}

monochromatic excitation at the frequency le

3. Specific excitation functions.

In this section the result of the above analysis ^is presented ^with particular ^{reference to a confocal} scanning ^acoustic microscope system, ^{for two different} excitation conditions

pulsed ^RF and linear FM

chirp pulses. ^{We assume} that the transmitter and

receiver lenses are identical and their point-spread

functions are adequately ^described by ^a J1(x)/x

type dependence. ^{We have :}

with

where D, r ^{and 1 are the} full lens aperture, ^radial distance, ^focal length respectively and v is the phase velocity in the medium. We further assume that the

frequency-dependent attenuation in the liquid ^can

be described by ^a square law function and ignore ^the

associated velocity dispersion ^which turns out to be very small at the frequencies of interest so that,

where c is a constant characterising ^the f 2 dependence

of attenuation.

The point spread ^{function at the centre of} gravity

of the spectrum ^{is then} simply given by :

with

In the following examples ^we concentrate on a specific

case, for which D ⁼ 150 _gm, 1 ⁼ _{100 ktm} and v = 1 500 ms-1 and the bandwidth is 500 MHz centred

on 750 MHz.

3.1 RF PULSE EXCITATION.

An RF pulse e(t)

centred at the frequency fo is given by :

where T is the pulsewidth. ^The corresponding ^Fourier

spectrum, E(t) ^is given by :

386

The effective point spread function, He ^calculated

form equation (8), ^is given by :

We have simulated the case where T ⁼ 20 ns for various loss conditions. Figure ^2a shows how the function T R varies in the ( f, r) plane for the lossless

case (i.e. ^{cl =} 0.0). Figure ^{2b shows} He ^calculated

from equation (17). This, within the approximations

we have made is the correct result It is interesting

to compare this with the point spread function cal- culated at the frequency fc

^{shown dotted in} figure ^2b.

In this, ^lossless case, spectral ^{centre of} gravity ^fre-

quency is identical to the band centre frequency

750 MHz. We see that using this intuitive approach

the main lobe is well predicted. ^{The main} departure

is in the « blurred » side lobe structure of the effective

point-spread function which is readily ^understood

to be due to a wideband signal.

Fig. ^2.

Point-spread function due to a 20 ns RF pulse centred on 750 MHz excitation with cl ⁼ 0.0. (a) ^variation

of TR in the ( f, r) plane ; (b) ^{variation of} He ^and He ^with

radial distance.

Figure 3a shows the surface when cl ⁼ 2.5 GHz- 2, corresponding ^{to a case where a 100 J.1m} focal length

lens focuses in water at room temperature. Figure ^3b

shows again the calculated He ^and Hc. ^{As far as the}

width of the main lobe is concerned we see that there is still good agreement ^between He ^and Hc. ^{The effect} of frequency averaging ^is again ^to smooth the sidelobe structure. Finally, figures 4a and b show the surface and He ^and Hc ^{for the case where cl = 10.0 GHz- 2.}

Although with such high losses the spectral ^centre

of gravity ^{is now} depressed ^{to 534 MHz we still see}

a remarkably good agreement between He ^and Hc.

Fig. ^3.

^As figure 2 but with cl ⁼ 2.5 GHz-2.

3.2 CHIRP PULSE EXCITATION.

A linear chirp pulse ^with bandwidth B and duration T can be written in the form,

where p ⁼ B/T ^{is the} dispersion ^{factor. It has} beçn

shown that for the case of large ^{BT the Fourier trans-}

form of e(t) ^is given by [13] :

Fig. ^4.

^As figure ^{2 but with cl = 10.0 GHz- 2.}

Which effectively ^{means constant} amplitude ^and quadratic phase ^{variation over the} passband ^The

schematic diagram ^{of a} pulse compression microscope using ^SAW filters and operating in reflection is shown in figure ^{5. We have} previously ^shown [14] ^{that in}

such a system ^{with both} expander ^and compressor

present ^the phase ^{terms cancel out} leading to,

Fig. ^5.

The basic elements of a pulse compression ^scan- ning ^acoustic microscope using surface acoustic wave

dispersive ^filters.

Figure ^{6a shows the TR} surface for the lossless

case. Figure ^{6b shows} He ^and He ^{for this case. As}

far as the main lobe is concemed a good agreement between He ^and He ^is observed Once again ^the He

sidelobe structure is much smoother than that asso-

ciated with monochromatic RF excitation. This is due to the fact that in this case all the frequencies

Fig. ^6.

^Point spread function due to a linear chirp pulse

centred on 750 MHz with 500 MHz bandwidth excitation with cl ⁼ 0.0. (a) variation of T R in the ( f, r) plane ; (b)

variation of He ^and He with radial distance.

contribute toward the effective point spread ^function equally. Figure ^{7a shows the case for cl = 2.5 GHz-2}

and figure 7b shows the corresponding He ^and Hc.

Again ^a good agreement for the mainlobe is evident.

It is seen that the sidelobe structure is in better _agree- ment with that of Hc(f). This is due to the fact that because of the large ^losses the effective spectrum ^is

narrower; ^{we are} closer to a monochromatic situation.

This effect is even more marked in figures ^{8a and b}

where the TR surface and He ^and Hc ^{have been} plotted

for cl ⁼ 10.0 GHz-2.

388

Fig. ^7.

^As figure 6 but with cl ⁼ 2.5 GHz- 2.

4. Conclusions.

We have presented ^an analytical theory ^{for the ana-} lysis of broadband imaging systems. ^The theory ^is général ; ^{it can be} applied ^to optical, acoustical and thermal imaging systems. ^{We have} illustrated our

results by ^{reference to a} specific ^case

^{that of} scanning ^acoustic microscopy. ^{We have shown that}

the performance ^{of a broadband} imaging system

can be well characterised by ^an equivalent ^mono-

Fig. ^8.

^As figure 6 but with cl ⁼ 10.0 GHz- 2.

chromatic system operating ^{at the} frequency ^corres- ponding ^{to the centre of} gravity ^{of the effective} power spectrum.

Acknowledgments.

The authors would like to express their gratitude

to Mr. M. Vaez Iravani for a number of fruitful discussions. The work was supported by the Science and Engineering ^{Research Council.}

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