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MEDT 8007 Linear field analysis
(Acoustic field from an ultrasound transducer)
Ingvild Kinn Ekroll
Dept. of Circulation and Medical Imaging
NTNU
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Linear field analysis?
• Applications of methods introduced in Cobbold’s chapter 2
• Chapter 3: How to calculate pressure fields from transducers (acoustic sources)!
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Lecture overview
• Brief sum up of chapter 2.1
• Reminder: The Rayleigh integral and diffraction
Linear field analysis
• Integral methods
• Impulse response methods
• Angular spectrum method
• Approximate methods
Field from a focused transducer
• Properties of the focused ultrasound beam
• Focusing the beam (short)
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Sum-up of chapter 2
• Derivation of Rayleigh-Sommerfeld eqs. and Rayleigh integral based on wave equation for velocity potential
• Green’s functions used to obtain solutions for the velocity potential given certain boundary conditions
– (surface velocity or pressure)
• The velocity potential formulation is mathematically convenient
– It is a scalar field for which the gradient gives the velocity vector field, whereas the temporal derivative gives the pressure
Velocity (vector) Pressure Pressure (monochromatic)
𝑣 𝑟, 𝑡 = −𝛻𝜑(𝑟, 𝑡) p(r, t) = ρ𝜕φ(r, t)
𝜕t 𝑃 𝑟, 𝜔 = 𝑖𝜔𝜌F(𝑟, 𝜔)
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The Rayleigh integral
𝜑 𝑟, 𝑡 = 1
2𝜋 𝑣𝑛 𝑡 − 𝑅/𝑐
𝑅𝑑𝑆
vn = Normal velocity component of the transducer surface
Velocity (vector) Pressure Pressure (monochromatic)
P 𝑟, 𝜔 =𝑖𝜌𝜔𝑣02𝜋 𝑒−𝑖𝜔
𝑅𝑐
𝑅𝑑𝑆
𝑣 𝑟, 𝑡 = −𝛻𝜑(𝑟, 𝑡) 𝑝(𝑟, 𝑡) = 𝜌𝜕𝜑(𝑟, 𝑡)
𝜕𝑡 𝑃 𝑟, 𝜔 = 𝑖𝜔𝜌F(𝑟, 𝜔)
Assumed single frequency excitation, vn = v0eiwt
S
S
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Diffraction
• Is what happens when a wave encounters a slit or obstacle
• Adding contributions at given locations from many point sources gives an interference pattern
Diffraction pattern!
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LINEAR FIELD ANALYSIS
Evaluating the Rayleigh equation
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Integral methods
• Direct numerical evaluation of Rayleigh integral
– Ex: Ultrasim (University of Oslo)
• More economical approaches?
– Assumptions: Plane transducer in rigid baffle + single frequency excitation
𝑃 𝑟, 𝑧, 𝜔 = 𝑖𝜔𝜌𝑣02𝜋 𝑒−𝑖𝑘𝑅
𝑅𝑑𝑆
Can be reduced to a line integral over the transducer periphery:
𝑃 𝑟, 𝑧, 𝜔 = 𝜌𝑐𝑣0𝑒−𝑖𝑘𝑧 −
𝜌𝑐𝑣02𝜋 𝑒−𝑖𝑘𝑅1(𝜃)2𝜋
0
𝑑𝜃
Plane wave part (in geometric shadow)
Edge/diffraction part everywhere
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What is a baffle?
• The baffle is the material surrounding the transducer surface
• Provides boundary conditions for the field response
• It is typically classified as either hard (stiff material, i), soft (ii), or pressure-release (iii, no pressure on bounding surface, e.g. air)
The material given around the transducer also play a part in determining the response at a given point in the field
S
Baffle
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Impulse response method
• Chapters 2.2 and 3.3 in Cobbold
• Reformulation of Rayleigh integral using the impulse response h(r,t) leads to:
𝜑 𝑟, 𝑡 = 𝑣𝑛 𝑡 ∗ ℎ(𝑟, 𝑡) 𝑝 𝑟, 𝑡 = 𝜌ℎ(𝑟, 𝑡) ∗𝜕𝑣𝑛(𝑡)
𝜕𝑡
• Impulse response? Velocity potential in observation point given d-exitation at all points of transducer surface
• Problem: We need to find h(r,t) in every spatial point
• Will be covered in upcoming lecture on Field II!
ℎ 𝑟, 𝑡 = 𝛿(𝑡 − 𝑅/𝑐)
2𝜋𝑅𝑑𝑆 𝜑 𝑟, 𝑡 =
1
2𝜋 𝑣𝑛 𝑡 − 𝑅/𝑐
𝑅𝑑𝑆
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Angular spectrum method
• Chapter 2.3 and 3.1 in Cobbold
• Frequency domain method
• Interesting from computational perspective due to effective implementations of 2D FFT
• The determined velocity field can be shown to equal the Rayleigh integral in the frequency domain
• Ex: FOCUS (University of Michigan)
• ASM will be covered in separate lecture!
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APPROXIMATIONS
From exact methods to
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Approximations – why?
• Quick predictions of radiation patterns (beyond near-field)
• Simpler expressions can provide clearer insight into important parameters that govern radiation patterns
• Provides «rule of thumb» also for focused beams and fields from arrays
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How?
x0 x
y0 y
Transducer surface, S Field point, P(x,y,z) |r-r0|
𝑅 = 𝑟 − 𝑟0 = 𝑧2 + (𝑥 − 𝑥0)2+(𝑦 − 𝑦0)2
≈ 𝑧 1 +1
2𝑧2𝑥2 + 𝑦2 − 2 𝑥𝑥0 + 𝑦𝑦0 + 𝑥02 + 𝑦02 + …
1)
P 𝑟, 𝜔 =𝑖𝜌𝜔𝑣02𝜋 𝐴𝑝𝑜𝑑(𝑥0, 𝑦0)𝑒
−𝑖𝜔𝑅𝑐
𝑅𝑑𝑆
P 𝑟, 𝜔 =𝑖𝜌𝜔𝑣02𝜋𝑅 𝐴𝑝𝑜𝑑(𝑥0, 𝑦0)𝑒
−𝑖𝜔𝑅𝑐𝑑𝑆
S
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• Mid- to far-field approximation
• vn(t) = Apod(x0,y0)v0eiwt (Surface velocity of transducer)
Fresnel approximation
𝑃(𝑥, 𝑦, 𝑧, 𝜔) = 𝑖𝜔𝜌𝑣0
2𝜋𝑅𝑒−𝑖𝑘 𝑧+
𝑥2+𝑦2
2𝑧 𝐴𝑝𝑜𝑑(𝑥0, 𝑦0)𝑒𝑖𝑘
𝑧𝑥0𝑥+𝑦0𝑦 −𝑥0
2−𝑦02𝑑𝑥0 𝑑𝑦0
• Given uniform excitation and piston transducer of radius a, the on-axis response is:
𝑃(0,0, 𝑧, 𝜔) = cρ𝑣0𝑒−𝑖𝑘𝑧(1 − 𝑒−𝑖𝑘𝑎
2/2𝑧)
S
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Fraunhofer approximation
x0 x
y0 y
Transducer surface, S Field point, P(x,y,z) |r-r0|
𝑅 = 𝑟 − 𝑟0 = 𝑧2 + (𝑥 − 𝑥0)2+(𝑦 − 𝑦0)2
≈ 𝑧 1 +1
2𝑧2𝑥2 + 𝑦2 − 2 𝑥𝑥0 + 𝑦𝑦0 + 𝑥02 + 𝑦02 + …
1)
P 𝑟, 𝜔 =𝑖𝜌𝜔𝑣02𝜋𝑅 𝐴𝑝𝑜𝑑(𝑥0, 𝑦0)𝑒
−𝑖𝜔𝑅𝑐𝑑𝑆
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• Far-field approximation (see ch 3.4 for more equations!)
• 𝑧 ≫𝜋(𝑥02+𝑦02)
l (or, less stringent 𝑧 ≫
𝐷2
2l)
• W is aperture function, enables full space integration
𝑃 𝑥, 𝑦, 𝑧, 𝜔 ≈𝑖𝜔𝜌𝑣02𝜋𝑅𝑒−𝑖𝑘 𝑧+
𝑥2+𝑦2
2𝑧 Á 𝐴𝑝𝑜𝑑W (𝑥, 𝑦)
Fraunhofer approximation
𝑃(𝑥, 𝑦, 𝑧, 𝜔) ≈ 𝑖𝜔𝜌𝑣0
2𝜋𝑅𝑒−𝑖𝑘 𝑧+
𝑥2+𝑦2
2𝑧 W(𝑥0, 𝑦0)𝐴𝑝𝑜𝑑(𝑥0, 𝑦0)𝑒𝑖𝑘
𝑧𝑥0𝑥+𝑦0𝑦 𝑑𝑥0 𝑑𝑦0
Resulting in the convenient formulation:
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Fraunhofer approximation
In words:
In the far field, the diffraction pattern from an ultrasound transducer is proportional to the Fourier transform of the aperture (times the apodization) function
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Accuracy of approximations
Far field
(Fraunhofer error <5%)
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Example – rectangular aperture
Fresnel approximation – lateral beam profile
• Assuming uniform excitation (Apod = 1)
• Valid only for single frequencies, but give important insight in diffraction/radiation patterns from any rectangular aperture
0 i /4 x xx
x L 2 x L 2p (x,z, ) e F F
2 z 2 z 2
-
w - l l
2z
i t /2
0
F(z) e dt-
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Example – rectangular aperture
x 0 i /4 xx
L L xp (x,z, ) e sinc
zz
w ll
Fraunhofer approximation (lateral beam profile) Similar results are given for the y-direction (elevation direction)
Fourier transform of a rectangle, i.e. flat apodization and finite aperture
• Assuming uniform excitation (Apod = 1) • Valid only for single frequencies, but give important
insight in diffraction/radiation patterns from any rectangular aperture
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Sound field characterization
Beam profile plots – 1-D / 2-D plot of axial and
lateral pressure
– Typically used to look at the beam width versus depth
Axial pressure plots – The pressure along the main
axis (x=0)
– Typically used to investigate depth penetration / uniformity
Fresnel
Fraunhofer
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Quick note on spatial resolution
• The ability of the imaging system to resolve two nearby objects
• The axial resolution (along the beam) is determined by the pulse length
• The lateral resolution is determined by the beam width
• In ultrasound imaging, the axial resolution is typically better (2-4x) than the lateral resolution
Lateral resolution
Axial resolution
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Quick note on contrast (resolution)
• The ability of an imaging system to discern (small) differences in scattering amplitude
• Mainly determined by:
– The beam side lobes
– Grating lobes (arrays)
– Acoustic beam distortion (beam aberration)
– Multiple reflections
Example: Liver tissue Carotid image courtesy of J. Rau
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Apodization
• Apodization results in reduced side lobes, but also widenes the main lobe. Reduces lateral resolution, increases contrast resolution.
• In conventional (array) imaging, apodization is mostly applied during reception (easier to implement)
No apodization -13 dB to first side lobe Hamming apodization -40.6 dB to first side lobe
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CW vs PW excitation
• Previously assumed: Continuous wave (CW)
• Pulsed wave (PW) excitation has finite length and a given bandwidth of frequency components
• How is the beam profile affected?
0 1 2 3 4 5 6 7 8 9 10-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Power
Frequency
Power
Frequency
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CW vs PW comparison
• CW excitation gives the ”expected” diffraction / interference pattern
• PW excitation has interference at different spatial locations for different frequency components, which smoothens the beam profile
CW beam profile PW beam profile
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CW vs PW comparison
• Axial pressure profile comparison
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Unfocused field calculation
Cardiac imaging – Aperture width D = 2 cm, transducer frequency f0 = 2.5 MHz,
sound speed c = 1540 m/s – Region of interest = 2-20 cm Far field, z_far = D^2/(2*lambda) = 32.4 cm (!) In other words: • Medical ultrasound imaging takes place in the near field of a
transducer • This would lead to a very wide (and complicated) beam in our
region of interest
How then do we get a well defined and narrow beam width in our region of interest?
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THE FOCUSED BEAM
Controlling the interference pattern
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Controlling the interference pattern
Two point sources: • Constructive and
destructive interference • Interference pattern
Multiple sources
• Constructive interference in one direction
Curved sources:
• Focused energy
• Constructive interference in a narrow area
One point source:
• Spherical waves
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Focusing the beam
• Focusing can be achieved by:
– Curving the transducer itself (previous slide)
– Using a lens in front of the transducer surface
– Electronically delaying the emitted signal on smaller elements in an array (coming up if there is enough time)
• Bringing the far-field into the near-field
– Accellerating the diffraction pattern
• In the focal region far-field relations hold
The beam pattern in the focal point is equal to the Fourier transform of the transducer aperture (times the apodization) function
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The focused transducer
Focal number (F-number/F#)
• The ratio of the focal depth to the aperture width
• Determines the lateral resolution (beam width)
– Lower F-numbers more narrow beam width
• Determines the depth of field (LF)
– Lower F-numbers shallow depth of field
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Depth-of-field
• The range of depths over which the beam width is approximately constant
Depth [cm]
Azim
uth
[cm
]
Beam profile - oneway/transmit/azimuth/RMS
0 0.5 1 1.5 2 2.5 3 3.5 4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-60
-50
-40
-30
-20
-10
0
2
F, 3dBL 7.2 Fnum- l
LF
The focused transducer
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Angular / lateral resolution
• Determined by the beam width
• In the focal point, we can use the Fourier transform of the aperture function to estimate the resolution
• Example: Rectangular aperture, flat apodization: – Fourier transform of rect sinc function
– First zero at F#*lambda = z/D*lambda
– This is also approximately the (full) -6dB beam width (power)
F#*lambda
~ -6dB beam width (power)
The focused transducer
Rectangular aperture
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Conventional beamforming Static Tx, dynamic Rx
Static Tx, static Rx
-5 5
15
50 [mm]
Dep
th [
mm
]
-5 5 [mm]
Focus
Poin
t scatterers