15.1 Introduction to Imaging

Converging and Diverging Lenses

• Converging lenses, thicker in the middle, converge light rays to form real images when rays cross, unless the object is at the focal point or nearer.

• Diverging lenses, thinner in the middle, diverge light rays, forming virtual images where rays appear to have come from.

Lens Properties and Image Formation

• Principal axis: an imaginary line passing through the lens center perpendicular to its surfaces.

• Focal point: where parallel rays converge or appear to diverge after passing through the lens.

• Focal length: the distance from the lens center to the focal point, determining the lens's strength.

• Optical power: measured in diopters (D), defined as the inverse of focal length (P = 1/f).

• Ray diagrams predict image formation assuming a thin lens and rays near the principal axis.

• Real images form where rays converge; virtual images form where rays appear to diverge.

Magnification and Special Cases

• Magnification calculated as the ratio of image height to object height (m = hi/ho = -v/u).

• Special cases include objects beyond the focal point forming real, inverted images, and objects at or closer than the focal point forming magnified, upright, virtual images.

• Magnifying glasses use converging lenses for magnification, producing virtual, upright images.

• The near point is the closest clear focus point for the normal eye, typically 25 cm, while the far point is the farthest clear focus point, typically at infinity.

Lens Aberrations and Mirrors

• Spherical aberration causes distorted images due to the lens's inability to focus all rays at the same point, which can be reduced by adapting the lens shape or using only the central lens portion.

• Chromatic aberration results from the lens's inability to bring rays of different colors to the same focus, which can be reduced by combining lenses with different shapes and refractive indices.

• Mirrors follow similar principles to lenses, with curved mirrors susceptible to spherical aberration.

• Telescope resolution depends on lens quality, aperture size, and wavelength, with improvements achieved through large apertures and small wavelengths.

Radio Telescopes

• Radio telescopes experience less atmospheric interference, and arrays using interferometry techniques can enhance resolution further.

15.2 Imaging Instrumentation

Microscope

Objective Lens and Eyepiece

• The objective lens forms a real magnified image of an object just beyond its focal point.

• The eyepiece acts as a magnifying glass to produce an inverted, magnified, and virtual final image.

Resolution and Magnification

• Resolution, the ability to distinguish separate points, is often more critical than magnification.

• Two objects are considered just resolvable if the angle they subtend is larger than 1.22λ/b (Rayleigh’s criterion), where b is the diameter of the receiving aperture.

Telescope

Objective Lens or Mirror

• The objective lens of a telescope forms a diminished, real, and inverted image of a distant object at its focal point.

• Reflecting telescopes use converging mirrors as their objectives.

Mounting Configurations

• Various mounting configurations exist, such as Newtonian and Cassegrain, each offering specific advantages.

Atmospheric Limitations and Solutions

• Optical telescopes on Earth's surface face limitations due to atmospheric effects.

• Placing telescopes on orbiting satellites overcomes these limitations.

Radio Telescopes and Interferometry

• Radio telescopes receive radio waves, which are less affected by the atmosphere.

• Interferometry techniques allow higher resolution by combining signals from multiple telescopes.

15.3 Fibre Optics

Data Transmission

• Data transmission occurs through electrical or infrared pulses in cables, with digital data represented by a large number of pulses, each having two possible levels.

• Transmission effects include attenuation, causing signal intensity loss with distance, and dispersion, resulting in pulse broadening and intensity decrease.

Fiber Optic Advantages and Principles

• Fiber optics offer advantages over copper, including lower attenuation, higher data rates, and immunity to electromagnetic interference.

• Total internal reflection ensures efficient wave propagation within fibers.

• Dispersion causes, such as waveguide and material dispersion, can be mitigated by using graded-index fibers or monochromatic light.

Additional Information

Attenuation Equation

• The intensity of a signal confined to an optic fiber decreases exponentially with distance along the cable (I = I0 * e^(-μx)).

15.4 Medical Imaging

X-ray Imaging

Absorption and Attenuation

• X-rays are absorbed and scattered differently by various body tissues, forming images based on these variations.

• The intensity of a parallel beam of X-rays decreases exponentially with distance due to absorption and scattering: I=I₀e^−μx , where μ is the linear attenuation coefficient.

• Absorption due to the photoelectric effect is a primary means of X-ray attenuation, dependent on the proton number (Z) of the atoms present.

Tissue Density and Attenuation

• Different tissue densities affect the intensity of the detected X-ray beam.

• The attenuation of X-rays can be characterized by the half-value thickness, representing the thickness of a medium that reduces transmitted intensity to half its previous value.

Image Quality Considerations

• X-ray quality is vital for high-quality images, balancing intensity, contrast, and safety.

Ultrasound Imaging

Wave Propagation and Reflection

• Ultrasound waves are directed into the body and reflect off boundaries between different tissues.

• Acoustic impedance determines the percentage of incident waves that reflect at tissue boundaries.

Pulse Transmission and Resolution

• Ultrasound waves are transmitted in pulses, with resolution improved by having several complete ultrasound waves in each pulse.

• Different ultrasound scan types, such as A-scans and B-scans, provide information about tissue position and size.

Frequency and Attenuation

• Higher ultrasound frequencies offer better resolution but also undergo more attenuation.

Magnetic Resonance Imaging (MRI)

Proton Spin Resonance

• MRI utilizes proton spin resonance in strong magnetic fields to produce images.

• Protons precess around the direction of the external magnetic field, with the rate of precession proportional to the field's strength.

Resonance and Imaging

• Resonance occurs when protons are subjected to an oscillating electromagnetic field of the same frequency.

• MRI provides three-dimensional images through gradient magnetic fields in three perpendicular directions.

Health Risks and Safety

Imaging Risks

• X-ray processes carry a health risk due to ionizing radiation exposure.

• MRI, not involving ionizing radiation, is considered safer than X-ray processes.

Additional Information

Attenuation Equation

• The intensity of a parallel beam of X-rays decreases exponentially with distance (I = I₀ e^{{-\mu x}}), where μ is the linear attenuation coefficient.

MRI Principles

• MRI utilizes proton spin resonance in strong magnetic fields to produce images.

• Resonance occurs when protons precess around the direction of the external magnetic field at the Larmor frequency, proportional to the field's strength.