Light as a Wave

Rays explain lenses, prisms, mirrors and the refraction of the eye. They cannot explain why a thin coating stops a lens reflecting, how an OCT scanner sees layers a few micrometres thick, why a tiny pupil blurs the image, how sunglasses cut glare, why a healthy cornea is clear, or why fluorescein glows. For those, light must be treated as what it is: a wave, and a stream of photons.

#Waves and photons

Light is an electromagnetic wave. Its wavelength λ and frequency f are linked by its speed, $c=f\lambda$. The eye responds to wavelengths from about 400 to 700 nm, from violet to red; shorter wavelengths are ultraviolet, longer ones infrared. In a medium of index n, light slows to c/n; its frequency, and so its colour, stays the same, and its wavelength shortens to λ/n.

Light is also absorbed and emitted in packets, photons, each carrying an energy

So a photon of blue or ultraviolet light carries more energy than one of red or infrared light, enough to drive damaging photochemical reactions: ultraviolet acts directly on proteins and DNA, and blue light mainly through photosensitisers such as lipofuscin that generate reactive oxygen. This is why short wavelengths cause photochemical damage, to the cornea, lens and retina, while infrared mainly heats tissue (Chapter 20). The retina’s sensitivity also depends on wavelength: in daylight it peaks near 555 nm, and in dim light, when rods take over, near 507 nm, so blues look relatively brighter at dusk (the Purkinje shift).

#Interference

Two waves that overlap add. Where crests meet crests the light is brighter (constructive interference); where crests meet troughs it cancels (destructive interference). For the pattern to stay still, the waves must keep a fixed phase relationship: they must be coherent, which in practice means they come from the same source by different paths. Waves whose paths differ by a whole number of wavelengths reinforce; by an odd number of half-wavelengths, they cancel.

Thin films show this everywhere: the colours of soap bubbles and oil slicks come from light reflected at the top and bottom of a film a few hundred nanometres thick. An anti-reflective coating uses it deliberately. A layer whose optical thickness is a quarter of a wavelength sends back a second reflection half a wavelength behind the first, and the two cancel. (Each reflection, from air to coating and from coating to the denser lens, also flips its phase by half a wave, but both flip, so only the extra path counts.) They cancel completely only if they are equally strong, which requires a coating index equal to the square root of the lens index:

No durable material has an index as low as √1.5 = 1.22, so a single magnesium fluoride layer (1.38) still leaves about 1.3 to 1.4% reflected at its design wavelength, and more toward the ends of the spectrum; multilayer coatings bring the average reflectance across the visible spectrum to about 0.5% or less per surface. The same physics makes interference filters, which pass or reflect narrow bands of wavelengths.

#Optical coherence tomography

OCT measures how deep each reflecting layer of tissue lies by interference. A Michelson interferometer splits light from a source into a reference beam, reflected by a mirror, and a sample beam, reflected by the layers of the retina or cornea. The returning beams combine and interfere, but only when their paths match within the source’s coherence length. A source with a broad spectrum has a very short coherence length, so interference picks out each layer’s depth precisely. The axial resolution depends on the source:

where λ₀ is the centre wavelength and Δλ the bandwidth: a broader bandwidth gives finer depth resolution. The lateral resolution is set by how finely the instrument focuses its beam, a separate matter of the optics and the eye’s pupil and aberrations.

• Time-domain OCT, the first kind, moved the reference mirror to scan through depth, point by point.
• Spectral-domain OCT keeps the mirror still and spreads the returning light across a spectrometer; a Fourier transform of the spectrum gives the whole depth profile at once, far faster. It typically uses light around 840 nm, with an axial resolution of about 5 µm.
• Swept-source OCT sweeps a tunable laser through its wavelengths, usually near 1050 nm. Longer wavelengths are scattered and absorbed less by the retinal pigment epithelium, so they reach deeper into the choroid, and swept-source systems lose less sensitivity with depth. (Spectral-domain OCT can image the choroid too, by placing it nearest the zero delay: enhanced depth imaging.) Anterior segment OCT uses about 1310 nm, which penetrates the sclera and angle structures; water absorbs it too strongly for it to cross the vitreous to the retina.
• OCT angiography repeats scans of the same place and maps where the signal changes between them: moving blood. It shows the retinal and choroidal circulation without dye, but not leakage, staining or pooling, and it misses very slow flow.
• Optical biometry uses the same low-coherence interferometry to measure the eye’s axial length, to the pigment epithelium (Chapter 16).

Interference also lets clinicians look past a cataract: two coherent beams passed through clear gaps in the lens form fringes on the retina, and the finest fringes the patient can see estimate the potential acuity of the retina (laser interferometry). The potential acuity meter does the same job without interference: it projects a tiny letter chart through a pinhole-sized aperture, aimed through a clear gap.

#Diffraction

A wave passing an edge or through an aperture spreads. Through a circular aperture, such as the pupil, a point of light is imaged not as a point but as a bright central disc surrounded by faint rings: the Airy pattern. The angle from the centre to the first dark ring is

and, by convention, two points closer than this are not resolved: the Rayleigh limit of resolution. At that separation the combined image dips by about a quarter between the two peaks; much closer and they blur into one. A smaller pupil means more diffraction. For a 2 mm pupil the limit is about 1.2′, close to the detail of a 20/20 letter, which is why pinholes much smaller than 1 mm make vision worse (Chapter 11), and why diffraction, not lens quality, limits the finest instruments.

Diffraction can also be put to work. A diffractive multifocal IOL carries fine concentric steps that send light into two or three diffraction orders at once, each a separate focus: for example, the zero order for distance and the first order for near (Chapter 16). Diffraction gratings spread white light into its spectrum, as in a spectrometer.

#Polarisation

Light waves vibrate across their direction of travel. Ordinary light vibrates in all directions at once; linearly polarised light vibrates in one plane. A polariser passes only the component along its axis: half of unpolarised light, and of polarised light a fraction given by Malus’s law,

where θ is the angle between the light’s plane of vibration and the polariser’s axis. Reflection polarises light. At Brewster’s angle, where $\tan {\theta }_B=n$, the reflected light is completely polarised parallel to the surface; for water, ${\theta }_B$ is 53°. Glare from a road or a lake is therefore mostly horizontally polarised, and sunglasses whose polarising axis is vertical remove it.

• Birefringent materials have different refractive indices for different planes of polarisation. The corneal stroma and the retinal nerve fibre layer are birefringent; scanning laser polarimetry measures the nerve fibre layer’s thickness from the change it makes in polarisation, after allowing for the cornea.
• Haidinger’s brushes, a faint yellow bow-tie seen when looking at a rotating polariser, arise because the macular pigment is dichroic and its molecules are aligned radially in Henle’s fibre layer, the photoreceptor axons around the fovea. They mark the fovea, so they can be used to locate and train foveal fixation.
• Polarised tests and displays present a different image to each eye through crossed polarisers: vectographic stereotests such as the Titmus fly (polarisers at 45° and 135°) and polarised binocular balance charts (Chapter 13). Most 3D cinemas use opposite circular polarisers instead.

#Scattering

Small particles scatter light out of a beam. Particles much smaller than the wavelength scatter in proportion to $1/{\lambda }^4$ (Rayleigh scattering), so blue light at 450 nm is scattered about six times as much as red at 700 nm: the sky is blue. Larger particles scatter all wavelengths more equally and mostly forward (Mie scattering): clouds are white, and the scattering of a cataract causes glare and veiling light. The slit lamp shows the light scattered back toward the examiner, but it is the forward scatter, onto the retina, that disables the patient: a posterior subcapsular cataract, near the nodal point, causes glare out of proportion to how it looks. A slit-lamp beam made visible by protein in the aqueous is the Tyndall effect: the flare of anterior uveitis.

#Absorption and fluorescence

Molecules absorb some wavelengths and not others. The ocular chromophores, melanin, haemoglobin, macular xanthophyll and water, each have their own absorption spectrum, which is why different lasers act on different tissues (Chapter 20). The cornea absorbs ultraviolet below about 295 nm, and the crystalline lens most of the rest of the ultraviolet up to about 400 nm; the lens yellows with age and absorbs more blue. Removing it restores the blue, and patients often notice that colours look bluer after cataract surgery (cyanopsia). Intraocular lenses absorb ultraviolet, and some are tinted to filter blue as well.

Some molecules re-emit absorbed energy as light of a longer wavelength within nanoseconds: fluorescence, and the shift to longer wavelengths is the Stokes shift. In angiography, an exciter filter passes only light the dye absorbs, and a barrier filter in front of the camera passes only the light it emits, so that only the dye is seen:

• Fluorescein absorbs blue light (peak about 490 nm) and emits yellow-green (about 525 nm). About 80% is bound to plasma proteins, chiefly albumin, and the free fraction fluoresces. It stays within normal retinal vessels, whose endothelial tight junctions form the inner blood–retinal barrier. It leaks freely from the fenestrated choriocapillaris, but the tight junctions of the retinal pigment epithelium, the outer barrier, keep it out of the retina.
• Indocyanine green absorbs and emits in the near infrared (about 805 and 835 nm). It is almost entirely protein-bound, so it stays in the choroidal vessels, and its long wavelengths pass through the pigment epithelium, haemorrhage and pigment: it images the choroid.
• Fundus autofluorescence needs no dye: lipofuscin in the retinal pigment epithelium fluoresces when excited by blue or green light, mapping the health of the pigment epithelium.

The filters must not overlap: if they do, blue light reflected from bright structures leaks through the barrier filter and mimics fluorescence (pseudofluorescence). True autofluorescence, from optic disc drusen or an astrocytic hamartoma, shows on photographs taken before the dye is injected. Fluorescence also makes the fluorescein pattern of a rigid contact lens visible (Chapter 15), and the cobalt blue light of the slit lamp is an exciter filter.

#Measuring light

Radiometry measures light as energy: power in watts, irradiance in watts per square metre. Photometry weights it by the eye’s sensitivity at each wavelength: luminous flux in lumens, luminous intensity (the flux a source sends into a unit solid angle) in candelas, illuminance (light falling on a surface) in lux, and luminance (light leaving it) in candelas per square metre. Illuminance from a point source is inversely proportional to the square of the distance, $E=I/d^2$. Bright sunlight gives about 100,000 lux, an office about 500; reading with low vision may need several times that. The light reaching the retina depends on the pupil too: retinal illuminance, in trolands, is the luminance multiplied by the pupil area in square millimetres. (Light entering near the edge of the pupil is less effective than light through its centre, because cones point toward the pupil centre: the Stiles–Crawford effect.)

Perimeters give luminance in apostilbs (1 asb = 1/π cd/m²). The Goldmann and Humphrey background is 31.5 asb (10 cd/m²), and a threshold in decibels is the attenuation from the brightest stimulus, 10,000 asb in the Humphrey: 0 dB is the brightest, and each 10 dB is ten times dimmer.

#Recap

• $c=f\lambda$; in a medium the wavelength shortens to λ/n. Photon energy $E=hc/\lambda$: short wavelengths are more damaging.
• Anti-reflective coating: optical thickness λ/4, ideal index √n of the lens.
• OCT: low-coherence interferometry. Axial resolution ∝ ${\lambda }_0^2/\Delta \lambda$ (broader bandwidth, finer); lateral resolution set by the focusing optics. Swept source near 1050 nm sees deeper.
• Airy disc: first dark ring at an angle of 1.22λ/D from the centre (the disc’s diameter is 2.44λ/D), the Rayleigh limit; small pupils diffract more.
• Malus: $I=I_0{\cos }^2\theta$. Brewster: $\tan {\theta }_B=n$; glare is horizontally polarised, so sunglasses pass vertical light.
• Rayleigh scattering ∝ 1/λ⁴. Tyndall effect: aqueous flare. Forward scatter causes glare. The cornea is clear because its thin fibrils are closely and evenly packed (order on a scale below λ/2).
• Fluorescein: absorbs about 490 nm, emits about 525 nm; blue exciter, yellow-green barrier filter. ICG: near infrared, protein-bound, images the choroid.

#Test yourself

#Further reading

• American Academy of Ophthalmology, Basic and Clinical Science Course, Section 3: Clinical Optics and Vision Rehabilitation, chapter on physical optics.
• E. Hecht, Optics. Pearson.
• D. Huang et al., Optical coherence tomography. Science 254 (1991) 1178–1181.
• D. M. Maurice, The structure and transparency of the cornea. Journal of Physiology 136 (1957) 263–286.