1TT Liu, BE280A, UCSD, Fall 2006Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2006Ultrasound Lecture 2TT Liu, BE280A, UCSD, Fall 2006Depth of Penetration! Assume system can handle L dB of loss, thenL = 20log10AzA0" # $ % & ' We also have the definition (= -1z20log10AzA0" # $ % & ' and the approximation(=(0fTotal range a wave can travel before attenuation L isz = L(0fDepth of penetration isdp=L2(0f2TT Liu, BE280A, UCSD, Fall 2006Depth of Penetration22041085133202401Depth ofPenetration(cm)Frequency(MHz)Assume L = 80 dB; α0= 1dB/cm/MHzTT Liu, BE280A, UCSD, Fall 2006Pulse Repetition and Frame Rate! Need to wait for echoes to return before transmitting new pulsePulse repetition interval isTR"2dpcPulse repetition rate isfR=1TRIf N pulses are required to form an image, then theframe rate isF =1NTR3TT Liu, BE280A, UCSD, Fall 2006Example! N = 256, L= 80dB, c = 1540m /s, "0= 1dB /cm / MHzWhat frequency should be used to achieve a frame rate of 15 frame/sec?TR=1FN= 0.26msTR#2dpc=L"0fcf #LacTR= 1.99MHzTT Liu, BE280A, UCSD, Fall 2006Huygen’s Principlehttp://www.fink.com/thesis/chapter2.htmlhttp://www.cbem.imperial.ac.uk/ardan/diff/hfw.html4TT Liu, BE280A, UCSD, Fall 2006Huygen’s PrincipleAnderson and Trahey 2000! Wavenumberk =2"#! Oliquity Factorr01TT Liu, BE280A, UCSD, Fall 2006Small-Angle (paraxial) ApproximationAnderson and Trahey 2000r015TT Liu, BE280A, UCSD, Fall 2006Fresnel ApproximationAnderson and Trahey 2000Approximates spherical wavefront with a parabolic phaseprofileTT Liu, BE280A, UCSD, Fall 2006Fresnel Approximation! U(x0, y0) =exp( jkz)j"z##expjk2zx1$ x0( )2+ y1$ y0( )2( )% & ' ( ) * s x1, y1( )dx1dy1=exp( jkz)j"zs(x0, y0) ++expjk2zx02+ y02( )% & ' ( ) * % & ' ( ) *6TT Liu, BE280A, UCSD, Fall 2006Fraunhofer Approximation! kr01" kz 1+12x1# x0z$ % & ' ( ) 2+12y1# y0z$ % & ' ( ) 2$ % & & ' ( ) ) = kz 1+12z2x12# 2x1x0+ x02( )+12z2y12# 2y1y0+ y02( )$ % & ' ( ) = kz +k2zx12+ y12( )+k2zx02+ y02( )#kzx1x0+ y1y0( )" kz + +k2zx02+ y02( )#kzx1x0+ y1y0( )Assume this termis negligible.TT Liu, BE280A, UCSD, Fall 2006Fraunhofer Condition! k2zx12+ y12( )Phase term due to position on transducer isFar-field condition is! k2zx12+ y12( )<< 1z >>k2x12+ y12( )="#x12+ y12( )For a square DxD transducer, x12+ y12= D2/4z >>"D24#$D2#7TT Liu, BE280A, UCSD, Fall 2006Fraunhofer ApproximationAnderson and Trahey 2000Quadratic phase termFourier transform of thesource with! kx=x0"z ky=y0"zTT Liu, BE280A, UCSD, Fall 2006Plane Wave (Fraunhofer) Approximationzx0x1θθdr01! d = "x1sin#sin#=x0r01$x0zd $ "x0x1z8TT Liu, BE280A, UCSD, Fall 2006Plane Wave Approximation! 1rexp( jkr) "1zexp jk z + d( )( )=1zexp j2#$z %x0x1z& ' ( ) * + & ' ( ) * + U(x0) = s(x1)%,,-1rexp( jkr)dx1 " s(x1)%,,-1zexp j2#$z %x0x1z& ' ( ) * + & ' ( ) * + dx1 =1zexp j2#z$& ' ( ) * + s(x1)%,,-exp %j2#x0x1$z& ' ( ) * + dx1 =1zexp j2#z$& ' ( ) * + s(x1)%,,-exp % j2#kxx1( )dx1 =1zexp j2#z$& ' ( ) * + F s(x)[ ]kx=x0$zTT Liu, BE280A, UCSD, Fall 2006Plane Wave Approximation! In generalU(x0, y0) =1zexp j2"z#$ % & ' ( ) F s(x, y)[ ]kx=x0#z,ky=y0#z,Examples(x, y) = rect(x / D)rect(y / D)U(x0, y0) =1zexp( jkz)D2sinc Dkx( )sinc Dky( ) =1zexp( jkz)D2sinc Dx0#z$ % & ' ( ) sinc Dkyy0#z$ % & ' ( ) Zeros occur at x0=n#zD and y0=n#zD Beamwidth of the sinc function is #zD9TT Liu, BE280A, UCSD, Fall 2006! "zD1! D1! "zD2! D22"! D12"! D2TT Liu, BE280A, UCSD, Fall 2006ExampleAnderson and Trahey 2000! rectxD" # $ % & ' rectxd" # $ % & ' (1dcombxd" # $ % & ' ) * + , - . / Dsinc(Dkx) ( d sinc(dkx)comb(dkx)[ ]SidelobesQuestion: What should we do to reduce the sidelobes?10TT Liu, BE280A, UCSD, Fall 2006Transducer DimensionAnderson and Trahey 2000! Goal : Operate in the Fresnel Zonez < D2/"Dopt#"zmaxExamplezmax= 20 cm"= 0.5 mmDopt= 1 cmTT Liu, BE280A, UCSD, Fall 2006Focusing in Fresnel Zone! U(x0, y0) =exp( jkz)j"z##expjk2zx1$ x0( )2+ y1$ y0( )2( )% & ' ( ) * s x1, y1( )dx1dy1=exp( jkz)j"z##expjk2zx12+ y12( )+ x02+ y02( )$ 2 x1x0+ y1y0( )( )% & ' ( ) * s x1, y1( )dx1dy1=exp( jkz)j"zexpjk2zx02+ y02( )% & ' ( ) * exp( jkz)j"z##expjk2zx12+ y12( )% & ' ( ) * exp $jkzx1x0+ y1y0( )% & ' ( ) * s x1, y1( )dx1dy1! Use time delays to compensate for this phase term! U(x0, y0) =exp( jkz)j"zexpjk2zx02+ y02( )# $ % & ' ( F expjk2zx12+ y12( )# $ % & ' ( s x1, y1( )) * + , - .11TT Liu, BE280A, UCSD, Fall 2006Focusing in Fresnel Zone! U(x0, y0) =exp( jkz)j"zexpjk2zx02+ y02( )# $ % & ' ( F expjk2zx12+ y12( )# $ % & ' ( s x1, y1( )) * + , - . Make ! s x1, y1( )= s0x1, y1( )exp "jk2z0x12+ y12( )# $ % & ' ( ! U(x0, y0) =exp( jkz0)j"z0expjk2z0x02+ y02( )# $ % & ' ( F s x1, y1( )[ ]At the focal depth z =z0Beamwidth at the focal depth is: ! "z0DTT Liu, BE280A, UCSD, Fall 2006Acoustic Lens! "z0D="FD c > c0z012TT Liu, BE280A, UCSD, Fall 2006Depth of Focus! When z " z0, the phase term is #$ = exp %jk2z0x12+ y12( )& ' ( ) * + exp %jk2zx12+ y12( )& ' ( ) * + and the lens is not perfectly focused. Consider variation in the x - direction.#$ =kx221z%1z0& ' ( ) * + For transducer of size D, x22=D24If we want #$ =,D22-1z%1z0& ' ( ) * + < 1 radian then1z%1z0<2-,D2The larger the D, the smaller the depth of
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