1TT Liu, BE280A, UCSD, Fall 2005Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2005Ultrasound Lecture 2TT Liu, BE280A, UCSD, Fall 2005Plane Wave (Fraunhofer) Approximationzx0x1θθdr01€ d = −x1sinθsinθ=x0r01≈x0zd ≈ −x0x1z2TT Liu, BE280A, UCSD, Fall 2005Plane Wave Approximation€ 1rexp( jkr) ≈1zexp jk z + d( )( )=1zexp j2πλz −x0x1z U(x0) = s(x1)−∞∞∫1rexp( jkr)dx1 ≈ s(x1)−∞∞∫1zexp j2πλz −x0x1z dx1 =1zexp j2πzλ s(x1)−∞∞∫exp −j2πx0x1λz dx1 =1zexp j2πzλ s(x1)−∞∞∫exp − j2πkxx1( )dx1 =1zexp j2πzλ F s(x)[ ]kx=x0λzTT Liu, BE280A, UCSD, Fall 2005Plane Wave Approximation€ In generalU(x0, y0) =1zexp j2πzλ F s(x, y)[ ]kx=x0λz,ky=y0λz,Examples(x, y) = rect(x /D)rect(y /D)U(x0, y0) =1zexp( jkz)D2sinc Dkx( )sinc Dky( ) =1zexp( jkz)D2sinc Dx0λz sinc Dkyy0λz Zeros occur at x0=nλzD and y0=nλzD Beamwidth of the sinc function is λzD3TT Liu, BE280A, UCSD, Fall 2005ExampleAnderson and Trahey 2000€ rectxD rectxd ∗1dcombxd ⇔ Dsinc(Dkx) ∗ d sinc(dkx)comb(dkx)[ ]SidelobesQuestion: What should we do to reduce the sidelobes? TT Liu, BE280A, UCSD, Fall 2005Huygen’s Principlehttp://www.fink.com/thesis/chapter2.htmlhttp://www.cbem.imperial.ac.uk/ardan/diff/hfw.html4TT Liu, BE280A, UCSD, Fall 2005Huygen’s PrincipleAnderson and Trahey 2000€ Wavenumberk =2πλ€ Oliquity FactorTT Liu, BE280A, UCSD, Fall 2005Small-Angle (paraxial) ApproximationAnderson and Trahey 20005TT Liu, BE280A, UCSD, Fall 2005Fresnel ApproximationAnderson and Trahey 2000Approximates spherical wavefront with a parabolic phaseprofileTT Liu, BE280A, UCSD, Fall 2005Fresnel Approximation€ U(x0, y0) =exp( jkz)jλz∫∫expjk2zx1− x0( )2+ y1− y0( )2( ) s x1, y1( )dx1dy1=exp( jkz)jλzs(x0, y0) ∗∗expjk2zx02+ y02( ) 6TT Liu, BE280A, UCSD, Fall 2005Fraunhofer Approximation€ kr01≈ kz 1 +12x1− x0z 2+12y1− y0z 2 = kz 1 +12z2x12− 2 x1x0+ x02( )+12z2y12− 2 y1y0+ y02( ) = kz +k2zx12+ y12( )+k2zx02+ y02( )−kzx1x0+ y1y0( )≈ kz + +k2zx02+ y02( )−kzx1x0+ y1y0( )Assume this termis negligible.TT Liu, BE280A, UCSD, Fall 2005Fraunhofer Condition€ k2zx12+ y12( )Phase term due to position on transducer isFar-field condition is€ k2zx12+ y12( )<< 1z >>k2x12+ y12( )=πλx12+ y12( )For a square DxD transducer, x12+ y12= D2/4z >>πD24λ≈D2λ7TT Liu, BE280A, UCSD, Fall 2005Fraunhofer ApproximationAnderson and Trahey 2000Quadratic phase termFourier transform of thesource with€ kx=x0λz ky=y0λzTT Liu, BE280A, UCSD, Fall 2005€ λzD1€ D1€ λzD2€ D12λz€ D22λz€ D28TT Liu, BE280A, UCSD, Fall 2005Transducer DimensionAnderson and Trahey 2000€ Goal : Operate in the Fresnel Zonez < D2/λDopt≈λzmaxExamplezmax= 20 cmλ= 0.5 mmDopt= 1 cmTT Liu, BE280A, UCSD, Fall 2005Focusing in Fresnel Zone€ U(x0, y0) =exp( jkz)jλz∫∫expjk2zx1− x0( )2+ y1− y0( )2( ) s x1, y1( )dx1dy1=exp( jkz)jλz∫∫expjk2zx12+ y12( )+ x02+ y02( )− 2 x1x0+ y1y0( )( ) s x1, y1( )dx1dy1=exp( jkz)jλzexpjk2zx02+ y02( ) exp( jkz )jλz∫∫expjk2zx12+ y12( ) exp −jkzx1x0+ y1y0( ) s x1, y1( )dx1dy1€ Use time delays to compensate for this phase term€ U(x0, y0) =exp( jkz)jλzexpjk2zx02+ y02( ) F expjk2zx12+ y12( ) s x1, y1( ) 9TT Liu, BE280A, UCSD, Fall 2005Focusing in Fresnel Zone€ U(x0, y0) =exp( jkz )jλzexpjk2zx02+ y02( ) F expjk2zx12+ y12( ) s x1, y1( ) Make € s x1, y1( )= s0x1, y1( )exp −jk2z0x12+ y12( ) € U(x0, y0) =exp( jkz0)jλz0expjk2z0x02+ y02( ) F s x1, y1( )[ ]At the focal depth z =z0Beamwidth at the focal depth is: € λz0DTT Liu, BE280A, UCSD, Fall 2005Acoustic Lens€ λz0D=λFD c > c0z010TT Liu, BE280A, UCSD, Fall 2005Depth of Focus€ When z ≠ z0, the phase term is ΔΦ = exp −jk2z0x12+ y12( ) exp −jk2zx12+ y12( ) and the lens is not perfectly focused. Consider variation in the x - direction.ΔΦ =kx221z−1z0 For transducer of size D, x22=D24If we want ΔΦ =πD22λ1z−1z0 < 1 radian then1z−1z0<2λπD2The larger the D, the smaller the depth of focus. TT Liu, BE280A, UCSD, Fall 2005Focusing with Phased ArrayAnderson and Trahey 200011TT Liu, BE280A, UCSD, Fall 2005Focusing and SteeringPrince and Links 2005TT Liu, BE280A, UCSD, Fall 2005Dynamic FocusingPrince and Links
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