Diffraction, polarization, lasers and OCT

4. Diffraction Limits, The Airy Disc & The Optimal Pupil Aperture

The Airy Disc

When light traverses the finite circular aperture of the human pupil, wave diffraction produces an interference pattern termed the Airy pattern, consisting of a central bright zone (the Airy disc) surrounded by concentric dark and bright rings. The central Airy disc contains $84\%$ of the total luminous energy.

The angular radius ($\theta$) from the central peak to the first dark minimum is given by the Fraunhofer diffraction formula:

$$\theta = \frac{1.22 \lambda}{D}\text{ radians}$$

where $\lambda$ is wavelength and $D$ is pupil aperture diameter.

Rayleigh's Criterion for Resolution

Under Rayleigh's criterion, two independent point sources are just resolvable when the central maximum of one Airy pattern falls directly onto the first minimum of the adjacent Airy pattern:

$$\theta_{\text{min}} = \frac{1.22 \lambda}{D}\text{ radians} = \frac{1.22 \lambda}{D} \times \left(\frac{180 \times 60}{\pi}\right)\text{ arcminutes}$$

For green light ($\lambda = 555\text{ nm}$) and a $2.5\text{ mm}$ pupil:

$$\theta_{\text{min}} = \frac{1.22 \times (555 \times 10^{-9})}{2.5 \times 10^{-3}} \approx 2.708 \times 10^{-4}\text{ rad} \approx 0.93\text{ arcminutes}$$

This $0.93$ arcminute resolution corresponds to the physical limit of standard $20/20$ Snellen visual acuity (which requires resolving $1.0$ arcminute detail).

The Ocular Pupil Trade-Off

Human visual performance represents a trade-off between two opposing optical phenomena:

1. Small Pupils ($< 2.0\text{ mm}$): Optical aberrations are reduced, but diffraction dominates. The Airy disc widens proportionally ($d \propto 1/D$), blurring retinal images.
2. Large Pupils ($> 4.0\text{ mm}$): Diffraction becomes negligible, but monochromatic higher-order aberrations dominate (spherical aberration, coma), degrading image quality.
3. Optimal Pupil Diameter: The optical compromise between diffraction and aberrations occurs at $2.5–3.0\text{ mm}$, often improving optical image quality; the optimum varies with aberrations, illumination and the task.

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5. Polarisation of Light in Ophthalmology

Light waves are transverse electromagnetic oscillations. In unpolarised light, the electric field vector oscillates randomly across all transverse planes. In linearly polarised light, oscillations occur in a single plane.

Brewster's Law

When unpolarised light strikes a non-metallic surface, reflected light becomes partially polarised. At a specific angle of incidence—Brewster's angle ($\theta_B$)—the reflected beam is $100\%$ linearly polarised perpendicular to the plane of incidence:

$$\tan \theta_B = \frac{n_2}{n_1}$$

For an air-to-water boundary ($n_1 = 1.000, n_2 = 1.333$): $\theta_B \approx 53.1^\circ$. For an air-to-cornea boundary ($n_1 = 1.000, n_2 = 1.376$): $\theta_B \approx 54.0^\circ$.

• Polarised Sunglasses: Contain polyvinyl alcohol-iodine filters with vertical transmission axes that block horizontally polarised reflective glare from roads or water.

Ocular Birefringence Applications

Anisotropic biological structures exhibit birefringence (differing refractive indices for perpendicular polarisation states):

• Cornea: Lamellar stromal collagen fibrils act as a birefringent crystal, introducing optical retardation.
• Retinal Nerve Fibre Layer (RNFL): Parallel cylindrical microtubules within unmyelinated retinal ganglion cell axons exhibit form birefringence. Scanning Laser Polarimetry (GDx) measures this phase retardation to quantify RNFL thickness for glaucoma management, employing a variable corneal compensator (VCC) to cancel out corneal birefringence.
• Henle's Fibre Layer: Radially oriented photoreceptor axons around the fovea exhibit dichroism and birefringence, generating the entoptic Haidinger's brushes phenomenon under plane-polarised blue light.

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6. Laser Optical Physics & Laser-Tissue Interactions

LASER is an acronym for Light Amplification by Stimulated Emission of Radiation.

Essential Physics of Laser Emission

1. Stimulated Emission: An incoming photon matching the energy transition $\Delta E = h\nu$ stimulates an excited electron to move to the lower lasing energy level, releasing an identical photon of the exact same wavelength, phase, direction, and polarisation.
2. Population Inversion: The lasing medium is pumped (via optical flashlamps, diodes, or electrical discharges) so that higher energy states have greater electron populations than the lower lasing levels ($N_2 > N_1$), reversing the thermal Boltzmann distribution.
3. Optical Resonator Cavity: Mirrors at opposite ends of the cavity (one $100\%$ high-reflector and one $95–99\%$ partially transmissive output coupler) feed photons back through the medium to amplify the coherent beam.
4. Laser Properties: Extreme monochromaticity, spatial and temporal coherence, and high collimation (low divergence).

Laser-Tissue Interaction Mechanisms

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7. Optical Coherence Tomography (OCT) Principles

Optical Coherence Tomography is non-invasive optical cross-sectional imaging based on low-coherence interferometry, analogous to ultrasound B-scan imaging but utilizing near-infrared light waves instead of acoustic waves.

Low-Coherence Interferometry (Michelson Interferometer)

Because light travels at $3.0 \times 10^8\text{ m/s}$, direct electronic time-of-flight measurements cannot achieve micron resolution. OCT solves this using a Michelson interferometer illuminated by a low-coherence superluminescent diode (SLD):

1. A $50:50$ beam splitter divides the light into two paths: a Sample Arm directed into the patient's eye and a Reference Arm directed towards a mirror.
2. Light backscattered from intraretinal layers recombines with light reflected from the reference mirror at the beam splitter.
3. Interference fringes occur only when the optical path length of the sample arm matches the reference arm to within the source's coherence length ($L_c$):

$$L_c = \frac{2 \ln 2}{\pi} \times \frac{\lambda_0^2}{\Delta \lambda}$$

Axial resolution depends principally on source centre wavelength and bandwidth, with tissue refractive index and dispersion also relevant. It is distinct from lateral resolution, which depends on the focusing optics and beam aperture. The displayed Gaussian-source relation illustrates the optical coherence-length scale; it does not make bandwidth the sole determinant of resolution in tissue.

Technological Generations: TD-OCT vs SD-OCT vs SS-OCT

• Time-Domain OCT (TD-OCT): Uses a mechanically moving reference mirror to sample depth point-by-point. Slow acquisition ($\approx 400$ A-scans/sec) with axial resolution $\approx 10\ \mu\text{m}$.
• Spectral-Domain OCT (SD-OCT / Fourier-Domain): The reference mirror is stationary. The recombined interference signal is dispersed by a diffraction grating spectrometer onto a high-speed linear CCD or CMOS detector. An inverse Fast Fourier Transform (FFT) extracts the entire depth profile (A-scan) simultaneously. Speeds reach $20,000–100,000+$ A-scans/sec with $3–5\ \mu\text{m}$ axial resolution and a $+20–30\text{ dB}$ sensitivity gain over TD-OCT.
• Swept-Source OCT (SS-OCT): Employs a rapidly tunable swept laser source centered at $\approx 1050\text{ nm}$ and a single high-speed photodetector. Speeds exceed $100,000–1,000,000+$ A-scans/sec. The longer wavelength ($1050\text{ nm}$) reduces light scattering by melanin and RPE, providing deep visualization of the choroid, choriocapillaris, and sclera.