Noll (2006) US Notes 2 page 1 Ultrasound Notes, Part II - Diffraction Analysis of the Lateral Response As discussed previously, the depth (z) response is largely determined by the envelop function, a(t). In the following sections, we will concern ourselves with deriving the beam pattern ),( zxxBz− . Consider the case of a focused US beam. How small of a spot can the beam be focused? Can it be infinitely small? What happens in depth planes other than the focal plane? What parameters control the resolution? The answers to these questions are given by a diffraction analysis. We will examine diffraction several different ways: 1. First, we consider a “steady-state” analysis, in which we ignore the envelope function. We will do this in both rectangular and polar coordinates. 2. We will then put the envelop function back into the equations and see its effect. 3. Finally, we go back to the “steady-state” analysis, but will consider an array transducer- that is, the transducer will be made of individual elements. Diffraction So, what is diffraction? – Historically, has meant optical phenomena that could not be explained by reflection (mirrors) or refraction (lenses). More generally, it has come to mean phenomena that can be caused by an interaction of wavefronts. The classic case is monochromatic light passing through two pinholes:Noll (2006) US Notes 2 page 2 Diffraction in Ultrasound: In these images, the transducer is indicated by the black line along the left margin and the transmitted wave is curved to focus at a particular point. Previously, we discussed the depth resolution was determined by the envelope function (in the case a Gaussian). The lateral localization function is more complicated and is determined by diffraction. Steady-State Diffraction in Ultrasound. Our model has the following characteristics: 1. Every point on the transducer (aperture) can be modeled using a spherical wave model, modified by a directional dependence term (cos θ) that corresponds to the transducer insertion efficiency. 2. We start our analysis by assuming a steady state model – that is, we will forget, for the moment, that this is a pulsed system and ignore the time propagation of both the pulse andNoll (2006) US Notes 2 page 3 the wavefronts. By ignoring time, we will, in essence, take a snapshot look at the wave fronts. 3. In this analysis, we will use the analytic signal (complex) representation for the true pressure wave, that is, we will use ikre instead of krcos , where ≡==λπω20ck wavenumber. The steady-state description of the pressure wave is then: θcos)(rerpikr= where θ is the obliquity angle. θa-aSourceAperturez 4. We will neglect attenuation and perform the analysis in 2D (ignoring the y dimension). a-azr0zxzx0 At some depth position z in the object and at a lateral position xz, the pressure signal will be the superposition of all point sources in the aperture. The superposition of sources of spherical wavefronts is known as the “Huygens-Fresnel” principle. The pressure wave functions is: 000cos),(0dxrexzpaazzikrzz∫−=θ, where ()2200zxxrzz+−= , zzrz00cos =θ. The variables x0 and xz are coordinates in the source plane and the plane at depth z, respectively. Simplifying a little, and we get: ()0200),( dxrzexzpaazikrzz∫−= (equivalent to first part of Macovski, 9.24)Noll (2006) US Notes 2 page 4 For any reflector at position (z, xz) (assume R=1), the pattern reflect to the transducer is exactly the same. This symmetrical relationship is known as the Helmholtz reciprocity theorem. The pressure wave at the source will be: () () ()20020200000),()(zikraazrikzikrzzrzexdrzerzexzpxpzzz′′==∫−′ The received signal can be represented as the integrated complex signal over the transducer: ()[]2202000),()()(0zaazikraazzzxzpdxrzedxxpxvz=⎥⎥⎦⎤⎢⎢⎣⎡==∫∫−− (equivalent to first part of Macovski, 9.25) Letting x be the position of the transmitter, the above equation represent the combined transmit/receive beam pattern for an ultrasound system: []2),(),(zzxxzpzxxB −=− Fresnel Zone. Whenever the depth of a reflector is substantially larger than the lateral displacement from all points in the source (transducer aperture):: ()202zxxz −>> then the Fresnel approximation holds. Specifically, the zikre0 term can be simplified using: ()()()()zxxzzxxzzxxzzxxrzzzzz202202202200212111−+=⎟⎟⎠⎞⎜⎜⎝⎛−+≅−+=+−= [ Here, we used the Taylor series expansion of uuf += 1)( expanded around u, keeping the first two terms uuuf2111)( +≅+=, where ()220zxxuz−= .]Noll (2006) US Notes 2 page 5 In addition to the usual (above) Fresnel approximation, we will simplify the smoothly varying scalar term (results are less sensitive to this approximations to this term): ()zrzz120≅ Applying both approximations, we get: ()0220),( dxezexzpaazxxikikzzz∫−−= Most ultrasound systems are assumed to operate in the Fresnel zone (where this approximation holds). Previously, p(z, xz) was the superposition of spherical wavefronts. With the Fresnel approximation, this is replaced with quadratic wavefronts. The make our solution more general, we now replace the source function with a potentially complex driving function: )(000)()(xexsxsφ= , which we will assume is bounded to [-a, a]. Our new pressure function for position (z, xz) is: ()⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡==∫∞∞−−zxikzikzzxxikikzzzzexszedxexszexzp2020220*)()(),( This says that the pressure pattern can be represented by the convolution of the driving function with a z-dependent, quadratic phase function (equivalent to Macovski, 9.31). This form is not particularly useful, but we can rewrite the pressure function so are more useful expression: 02022002)(),( dxeexszeexzpzxikzxxikzxikikzzzz∫∞∞−−=Noll (2006) US Notes 2 page 6 Fraunhoffer Zone. Even farther away from the transducer, another approximation can be made. For the “Fraunhoffer approximation,” we require that: π<<zkx220 for all ],[0aax−∈, or π<<zka22 or πλπ<<za222 or λ2az >> , usually we use λ24az > Under these assumptions, 1220≈zxike 00202)(),( dxexszeexzpzxxikzxikikzzzz∫−= With a simple substitution, zxuzλ= , we get: {}zxuzxikikzzxikikzzzzuxizxszeedxexszeexzpλπ=ℑ==∫−)()(),(020022022 (equivalent to Macovski, 9.39) That is, the field pattern in the Fraunhoffer zone (often call “far field”), is