Prisms

A prism is a block of glass or plastic with two flat faces set at an angle. Because its faces are flat it cannot focus light the way a lens does. What it can do is steer light: every ray that passes through it is turned through nearly the same angle. Clinicians use that steering to measure misaligned eyes and to relieve double vision, and patients meet it, whether anyone intends it or not, every time they look through a spectacle lens away from its centre.

#A wedge of glass

The thin edge of a prism is its apex and the thick edge its base. The angle between the two faces is the apex angle, $A$. A ray passing through is refracted twice, once as it enters and once as it leaves. Change the apex angle with the slider:

Both refractions turn the ray the same way: toward the base. Entering the glass, the ray bends toward the normal of the first face (grey dashed lines mark the normals), and because that face leans back toward the base, bending toward its normal means bending toward the base. Leaving the glass, the ray bends away from the normal of the second face, which leans the other way, so again it turns toward the base.

The total angle through which the ray turns is the deviation, δ. A thicker wedge, with a larger apex angle, deviates light more. For the thin prisms used in clinical work the relationship is simple:

For glass, with $n\approx 1.5$, a prism deviates light by about half its apex angle. As the readout under the figure shows, the approximation drifts further from the exact value as the prism gets stronger.

#The prism dioptre

Clinicians rarely measure deviation in degrees. They use the prism dioptre, written Δ. A prism of one prism dioptre deviates light by one centimetre at a distance of one metre:

A 10 Δ prism moves the ray 10 cm on a screen a metre away; a 5 Δ prism moves it 5 cm. For small angles one prism dioptre is about 0.57°, or one degree is about 1.75 Δ. At large angles the two scales part company: a deviation of 45° is 100 Δ, not 79 Δ.

#Images move toward the apex

What does a prism do to what you see? An eye looking through a prism receives light that has been bent toward the base. The eye has no way of knowing that the light was bent, so it sees the object in the direction the light finally arrives from.

The object is seen displaced toward the apex. A base-down prism lifts the world up and a base-up prism lowers it, and to look at the object the eye has to turn toward the apex too. An easy way to remember it: the light goes to the base, the image goes to the apex.

#Prisms for squint

In strabismus, or squint, the two eyes do not point at the same object. Suppose the right eye turns in toward the nose, an esotropia. Light from whatever the left eye is looking at then lands not on the right eye’s fovea but beside it, on its nasal retina, and the patient may see double.

The figure shows a right esotropia of 20 Δ, seen from above. Without a prism, the image in the right eye lands about 3.4 mm from its fovea. The brain treats the nasal retina as looking out to the temporal side, so the second image appears on the right: uncrossed double vision, typical of esotropia. Now add a base-out prism in front of the right eye and increase its power. The prism bends the light toward its base, outward, until at 20 Δ the light arrives along the turned eye’s line of sight and falls on the fovea. The double vision disappears.

Two rules follow from this.

• Put the apex of the prism toward the deviation, the direction the eye is turned: base out for an esotropia (an eye turned in), base in for an exotropia (an eye turned out), base down in front of an eye that is turned up, base up in front of one that is turned down.
• The prism that exactly neutralises the deviation measures it. Increasing prism is held in front of an eye during the alternate cover test until the eyes no longer move to take up fixation; that power, in prism dioptres, is the size of the squint.

#How you hold a prism matters

So far the light has always met the prism in the same way. Tilt the prism and its deviation changes. In the clinic, where loose prisms are held up in front of patients’ eyes by hand, that matters more than you might expect. The figure follows a prism labelled 40 Δ as its angle to the light changes:

The curve has a minimum. When the light passes through symmetrically, entering and leaving at the same angle, the deviation is as small as it can be. Tilt the prism either way from there and the deviation grows, slowly at first and then quickly.

Every prism is therefore calibrated for one particular position, and delivers its labelled power only when held that way:

• Glass prisms are calibrated in the Prentice position, with the back face perpendicular to the line of sight of the eye looking through it, so that all of the bending happens at the front face.
• Plastic prisms, including prism bars, are calibrated for minimum deviation. In practice they are held in the frontal-plane position, with the back face parallel to the patient’s face, which is close to minimum deviation.

Weak prisms forgive a careless position; strong ones don’t. Held in the frontal plane, a 40 Δ glass prism delivers only about 32 Δ. Tilted into the Prentice position, a 40 Δ plastic prism delivers over 70 Δ. Nor should two prisms be stacked with their bases the same way in front of one eye. Prism dioptres measure the tangent of the deviation, so they do not add at large angles even when the angles do, and the second prism receives light that has already been deviated, away from its calibration position: the result can be much larger than the sum of the two labels. A horizontal and a vertical prism may be combined, or the power can be split between the two eyes.

#Colour fringes

A prism also separates colours, because the refractive index, and with it the deviation, is a little larger for blue light than for red.

Each of these prisms deviates yellow light by the same 30 Δ; the higher the refractive index, the smaller the apex angle it needs. With crown glass, whose Abbe number is 59, the spectrum a metre away is about 8 mm wide. Polycarbonate, with an Abbe number of 30, spreads it twice as wide. Patients notice this as coloured fringes around objects when they look through strong prisms, or through the edges of strong lenses, which, as we’ll see in the next chapter, act as prisms too.

#Prisms add like arrows

A prism’s effect has a size and a direction, the direction of its base, so weak prisms combine like arrows. A prism of 3 Δ base in and one of 4 Δ base up act together, very nearly, as a single prism of $\sqrt{3^2+4^2}=5\Delta$, with its base pointing between the two.

The Risley prism built into a phoropter puts this to work. It holds two equal prisms that rotate in opposite directions. When their bases point the same way their powers add; when they point in opposite directions they cancel; in between, the resultant changes smoothly.

Two 10 Δ prisms give anything from 0 to 20 Δ, always along the same line. Clinicians use Risley prisms to measure phorias (latent deviations) and fusional reserves.

#Fresnel prisms

A strong prism is thick and heavy, yet only the angle between its faces matters, not the glass in between. Cut the prism into narrow horizontal strips, keep the face angle of each strip, and line them up against a thin back:

Each groove deviates light just as the solid prism did, but the whole thing is now only about a millimetre thick. This is a Fresnel prism. Made of flexible plastic, Fresnel prisms are pressed onto the back of a spectacle lens, which makes them ideal for trying out a prism or for a deviation that is expected to change. The price is some blur and loss of contrast from the grooves, both of which increase with the power.

#Recap

• A prism deviates light toward its base; things seen through it appear displaced toward its apex.
• For a thin prism $\delta \approx (n-1)A$. Prism power in prism dioptres is $\Delta =100\tan \delta$: 1 cm at 1 m.
• For strabismus, put the apex toward the deviation: base out for esotropia, base in for exotropia.
• The deviation depends on how the prism is held. Glass prisms are calibrated in the Prentice position, plastic prisms for minimum deviation (hold them in the frontal plane).
• Weak prisms add, very nearly, as vectors. Risley prisms vary smoothly from zero to the sum of two prisms; Fresnel prisms are thin and temporary.

#Test yourself

#Further reading

• J. T. Thompson and D. L. Guyton, Ophthalmic prisms: measurement errors and how to minimize them. Ophthalmology 90 (1983) 204–210.
• M. Jalie, Ophthalmic Lenses and Dispensing. Butterworth-Heinemann.
• American Academy of Ophthalmology, Basic and Clinical Science Course, Section 3: Clinical Optics and Vision Rehabilitation.