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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
2
Linear field analysis?
• Applications of methods introduced in Cobbold’s chapter 2
• Chapter 3: How to calculate pressure fields from transducers (acoustic sources)!
3
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)
4
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)
5
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
6
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!
LINEAR FIELD ANALYSIS
Evaluating the Rayleigh equation
8
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
9
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
10
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
11
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!
APPROXIMATIONS
From exact methods to
13
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
14
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
15
• 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
16
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
17
• 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:
18
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
19
Accuracy of approximations
Far field
(Fraunhofer error <5%)
20
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-
21
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
22
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
23
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
24
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
25
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
26
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
27
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
28
CW vs PW comparison
• Axial pressure profile comparison
29
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?
THE FOCUSED BEAM
Controlling the interference pattern
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
32
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
33
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
34
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
35
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
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