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This lesson gathers the non-ionising imaging modalities under AQA spec 3.10.4 that depend on the propagation, reflection and guiding of waves rather than on ionising radiation. Two physical themes hold the topic together: acoustic-impedance mismatch (the key parameter governing reflection in diagnostic ultrasound) and total internal reflection (the key principle behind fibre optics and endoscopy). The diagnostic radiology corridor and the operating theatre both rely heavily on these techniques because they avoid the cumulative dose of X-rays and the contraindication list of MRI — but they bring their own resolution and depth-of-field constraints, which this lesson develops.
Spec mapping: This lesson sits under AQA 7408 section 3.10.4 (Non-ionising imaging — ultrasound, fibre optics and endoscopy). It covers the ultrasound frequency range used in medicine (typically 1-15 MHz, with deep-organ imaging using lower frequencies and ophthalmic and superficial work using higher frequencies); generation and detection by piezoelectric transducer materials such as PZT (lead zirconate titanate) and PVDF (polyvinylidene fluoride); acoustic impedance Z = ρc; the intensity reflection coefficient α_r = (Z_2 - Z_1)²/(Z_2 + Z_1)² and the necessity of a coupling gel for impedance matching at the skin; A-scan vs B-scan imaging modes; Doppler ultrasound for blood-flow measurement; fibre optics governed by total internal reflection, with step-index and graded-index designs and monomode vs multimode fibres; the optical structure of an endoscope (illuminating bundle and coherent imaging bundle) and the resolution limit set by single-fibre core diameter. (Refer to the official AQA specification document for exact wording.)
Synoptic links:
- Section 3.4 (waves): ultrasound is a longitudinal mechanical wave; the reflection coefficient is the acoustic analogue of the impedance-mismatch formula for waves on a string with joined sections. Doppler ultrasound applies the Doppler effect from the same waves topic.
- Section 3.3.2 (refraction and total internal reflection): fibre optics is total internal reflection in a long, thin geometry; the critical-angle formula sin θ_c = n_2/n_1 governs the acceptance angle and numerical aperture.
- Section 3.10.2 (physics of the ear): the same impedance-matching argument that explains why the ossicles boost coupling between air and cochlear fluid explains why ultrasound gel is essential between transducer and skin.
Medical ultrasound uses sound at frequencies far above the human upper hearing limit (about 20 kHz). The diagnostic range is typically 1-15 MHz, with the choice of frequency a trade-off between two competing effects:
The frequency chosen therefore depends on the depth of the target:
| Application | Typical frequency | Typical penetration |
|---|---|---|
| Obstetric (foetal) imaging | 3-7 MHz | 10-20 cm |
| Abdominal imaging | 2-5 MHz | up to 25 cm |
| Echocardiography (cardiac) | 2-5 MHz | up to 20 cm |
| Vascular (peripheral) | 5-10 MHz | up to 10 cm |
| Ophthalmic / superficial | 10-15 MHz | a few cm |
The values are textbook ranges; clinical practice tunes them to patient body habitus and the specific task.
Ultrasound transducers exploit the piezoelectric effect: certain crystalline materials produce a voltage when mechanically deformed (direct effect) and conversely deform when a voltage is applied (inverse effect). Standard transducer materials are lead zirconate titanate (PZT) — a polycrystalline ceramic with high coupling efficiency — and polyvinylidene fluoride (PVDF) — a polymer with low impedance closer to tissue, useful in some specialist applications.
The transducer is built as a thin disc of piezoelectric material with electrodes on both faces. Driven with an electrical pulse, it produces a short ultrasonic pulse; switched to receive mode, it produces a voltage signal in response to returning echoes. The same element therefore both transmits and receives — the timing of the round trip gives the depth, and the amplitude gives the reflectivity.
For maximum efficiency, the thickness of the piezoelectric disc is chosen to be λ/2 at the operating frequency (where λ is the wavelength of ultrasound in the piezoelectric material). The disc then resonates at the operating frequency, acting as a half-wave acoustic cavity.
The acoustic impedance of a medium is
Z = ρc
where ρ is density and c is the speed of sound in that medium. Z has units of kg m⁻² s⁻¹ (or rayls).
| Medium | ρ (kg m⁻³) | c (m s⁻¹) | Z (kg m⁻² s⁻¹) |
|---|---|---|---|
| Air | 1.2 | 340 | ~410 |
| Water | 1000 | 1480 | 1.48 × 10⁶ |
| Soft tissue (avg) | 1060 | 1540 | 1.63 × 10⁶ |
| Fat | 950 | 1450 | 1.38 × 10⁶ |
| Muscle | 1075 | 1590 | 1.71 × 10⁶ |
| Bone | 1900 | 4000 | 7.6 × 10⁶ |
| PZT (ceramic) | 7500 | 4000 | 3.0 × 10⁷ |
The fractional reflected intensity at a boundary between media of impedance Z_1 and Z_2 is
α_r = (Z_2 - Z_1)² / (Z_2 + Z_1)²
Two key consequences:
Calculate the fractional reflection of ultrasound at an air-soft-tissue interface, and compare with a gel-soft-tissue interface (Z_gel ≈ 1.5 × 10⁶ kg m⁻² s⁻¹).
Air to soft tissue: α_r = ((1.63 × 10⁶) - 410)² / ((1.63 × 10⁶) + 410)² ≈ (1.63 × 10⁶ / 1.63 × 10⁶)² → essentially 1 (≈ 99.9% reflection). Almost nothing penetrates.
Gel to soft tissue: α_r = ((1.63 × 10⁶) - (1.50 × 10⁶))² / ((1.63 × 10⁶) + (1.50 × 10⁶))² = (1.3 × 10⁵)² / (3.13 × 10⁶)² = 1.69 × 10¹⁰ / 9.80 × 10¹² ≈ 0.0017 or about 0.17% reflection.
Without gel, virtually no ultrasound reaches the patient; with gel, virtually all of it does. The gel is not a comfort feature — it is the imaging-system enabler.
The simplest ultrasound display. A pulse is transmitted; echoes are recorded over time and plotted as amplitude vs time (equivalently, vs depth, with depth d = ct/2 for the round trip). The result is a one-dimensional trace with peaks at each tissue boundary. A-scan is mostly historical now but survives in ophthalmic biometry — measuring axial eye length pre-cataract surgery — where its high distance precision is valuable.
The dominant clinical mode. A transducer array sweeps a beam across a 2-D plane through the patient; at each beam direction, the time-amplitude trace is converted into a line of pixels whose brightness encodes echo amplitude. Combining many lines builds a 2-D image. The familiar grey-scale obstetric and abdominal images are B-scans.
When ultrasound reflects off a moving target — most importantly, red blood cells in a vessel — the returning frequency is shifted. The Doppler shift is
Δf = (2 f_0 v cos θ) / c
where f_0 is the transmitted frequency, v is the target speed, c is the sound speed in tissue, and θ is the angle between the beam and the velocity vector. The factor of 2 reflects the fact that the moving scatterer is both a moving receiver and a moving source. Vascular Doppler measures blood-flow velocities (typically 0.1-1 m s⁻¹) and is the workhorse of vascular surgery and obstetric foetal-heart assessment.
A 5 MHz Doppler probe is aligned at 60° to a blood vessel. The measured Doppler shift is 1.62 kHz. Calculate the blood-flow speed.
v = (Δf × c) / (2 f_0 cos θ) = (1620 × 1540) / (2 × 5 × 10⁶ × 0.500) = (2.49 × 10⁶) / (5.00 × 10⁶) = 0.50 m s⁻¹.
The result is a typical arterial flow velocity. Note the strong dependence on θ — at θ = 90° the Doppler shift vanishes, which is why probe angulation is the operator's principal control.
An optical fibre is a thin filament — typically 5-100 μm in core diameter — of high refractive-index material (n_1) surrounded by a cladding of lower refractive index (n_2). Light injected at one end at an angle less than the acceptance angle undergoes total internal reflection at every encounter with the core-cladding boundary and emerges from the other end.
The condition for total internal reflection at the core-cladding boundary is θ > θ_c, where
sin θ_c = n_2 / n_1.
For a typical silica core with n_1 = 1.50 and cladding n_2 = 1.48, θ_c = arcsin(1.48/1.50) = 80.6°. This is large — the ray must skim the wall at a glancing angle — but the geometry of injection from outside the fibre into the core ensures that any ray within the numerical aperture of the fibre satisfies the condition. Typical numerical aperture is 0.2-0.5, corresponding to acceptance half-angles of 12-30°.
Step-index fibre — the refractive index is constant in the core and steps to a lower value in the cladding. Simple and cheap, but rays at different angles travel different path lengths and arrive at different times — modal dispersion.
Graded-index fibre — the refractive index varies smoothly from a maximum at the core centre to the cladding value at the edge. Rays at steeper angles travel further but through lower-index material (faster), partially compensating for path-length differences. Reduced modal dispersion.
Multimode fibre — core diameter typically 50-100 μm. Many transverse modes propagate. High light-gathering, easy to couple to. Used in short-range applications including most medical endoscopes.
Monomode (single-mode) fibre — core diameter ≈ 9 μm. Only one transverse mode propagates. Eliminates intermodal dispersion. Used in long-distance telecom and in some specialist medical imaging (e.g. optical coherence tomography).
An endoscope inserts an optical pathway into a body cavity — gastrointestinal, urinary, respiratory, joint — to permit direct visualisation, biopsy, and minimally invasive surgery. The basic optical structure has two functional bundles:
Illuminating bundle — a non-coherent bundle of fibres carrying light from an external source to the distal tip. Order of fibres does not matter; the only requirement is that they deliver light.
Imaging bundle (coherent bundle) — a tightly ordered bundle of fibres in which the spatial position of each fibre at the distal (sample) end is mirrored exactly at the proximal (eyepiece / camera) end. Each fibre carries one "pixel" of the image. Disordering the bundle scrambles the image.
The lateral resolution of a fibre-optic endoscope is limited by the core diameter of a single imaging-bundle fibre — typically 5-10 μm. A clinical bundle has 10,000-50,000 fibres in a few millimetres diameter.
Modern endoscopes have largely replaced fibre-optic imaging bundles with miniature CCD or CMOS image sensors at the distal tip ("videoscopes"), but the optical-fibre principle remains in flexible endoscopes used in many specialties and in capsule endoscopy.
graph LR
A["Light source<br/>(xenon arc / LED)"] --> B["Illuminating<br/>fibre bundle"]
B --> C["Distal tip<br/>in body cavity"]
C --> D["Reflected light<br/>from tissue"]
D --> E["Coherent imaging<br/>bundle (~10k fibres)"]
E --> F["Eyepiece /<br/>camera"]
F --> G["Display"]
style B fill:#f39c12,color:#fff
style E fill:#27ae60,color:#fff
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