Astigmatism

Every lens so far has looked the same from every side. Turn a spherical lens about its centre and nothing changes: its surfaces are pieces of spheres, curved equally across every diameter. But a surface can curve more steeply one way than the other, like the side of a barrel or the back of a spoon. Most corneas are a little like that, and so are the lenses that correct them.

#Cylinders

The extreme case is a lens cut from a glass cylinder. Along the length of the cylinder its surface is straight; across it, the surface is curved. Light crossing the lens along the straight direction meets what is, in effect, a flat plate and goes straight on. Light crossing it the other way meets a lens. So a cylindrical lens focuses light in one direction only, and instead of converging to a point, parallel light is squeezed into a line:

The straight direction is the cylinder’s axis. Light spread out along the axis stays spread out, while light spread across it is brought together, so the focal line always lies parallel to the axis. This cylinder has a power of +2.00 D across its axis and none along it, so its focal line forms 50 cm behind the lens. Turn the axis and the line turns with it. You can also drag the figure to look at it from other angles.

#Naming directions

To prescribe a cylinder we need to say which way its axis points. Clinicians give it as an angle, measured anticlockwise from the horizontal as the examiner looks at the patient’s face. The vertical is 90, and the horizontal is called 180 rather than 0. This is the standard (or TABO) notation, and it is the same for both eyes. The axis follows a multiplication sign, so the cylinder above, with its axis horizontal, is written

and read “plano, plus two, axis one-eighty”. The first number is the sphere, none here, and the second is the cylinder. The axis is written with three digits, 090 rather than 90, and without a degree sign, because a hastily written ° is easily read as an extra zero.

#Power in between

Any line through the centre of a lens is a meridian, named by its angle in the same way. A cylinder has no power in the meridian along its axis and its full power in the meridian at right angles to it. What about the meridians in between? Drag the meridian around this +2.00 D cylinder, whose axis is vertical:

The curvature, and with it the power, grows smoothly from nothing along the axis to its full value at right angles to it:

where $C$ is the power of the cylinder, $\alpha$ its axis and $\theta$ the meridian. Thirty degrees from the axis, as at the meridian of 060 where the figure starts, a cylinder has a quarter of its power; at 45° it has half.

#Spherocylinders

Most astigmatic prescriptions combine a sphere and a cylinder. Take +1.00 / +2.00 × 090. The sphere adds +1.00 D in every meridian. The cylinder adds nothing along its axis, at 90, and +2.00 D across it, at 180. So the lens has +1.00 D in its vertical meridian and +3.00 D in its horizontal one. Every such lens has two principal meridians at right angles, one with the power of the sphere, $S$, and the other with $S+C$. Clinicians sketch them as a power cross:

Now press Transpose. The powers stay where they were, +1.00 D vertically and +3.00 D horizontally, but the prescription changes: +1.00 / +2.00 × 090 becomes +3.00 / −2.00 × 180. Only the colours swap, because blue always marks the meridian whose power is the sphere. Both prescriptions describe the same lens. One starts from the vertical meridian’s +1.00 D and adds 2.00 D horizontally; the other starts from the horizontal meridian’s +3.00 D and takes 2.00 D away vertically. To transpose a prescription:

• add the cylinder to the sphere to get the new sphere,
• change the sign of the cylinder,
• and turn the axis through 90°.

Most optometrists write prescriptions with minus cylinders, while many ophthalmologists use plus cylinders. Both are correct, as long as you can move between them.

One more number will matter shortly: the average of the two principal powers, the spherical equivalent.

For +1.00 / +2.00 × 090 it is +2.00 D, and it is the same whichever form the prescription is written in.

#Two focal lines

What does a spherocylindrical lens do to a beam of parallel light? Each principal meridian behaves as a lens of its own. The horizontal meridian of +1.00 / +2.00 × 090 has +3.00 D, so it brings light together 33.3 cm behind the lens. The vertical meridian has only +1.00 D and takes a full metre. Move the screen through the beam and watch its cross-section:

Just behind the lens the beam is round. As it travels it narrows faster horizontally than vertically, becoming an upright ellipse. At 33.3 cm the horizontal meridian comes to a focus and the beam collapses into a vertical line. Beyond it the light spreads out again horizontally while it carries on converging vertically, until at 1 m the vertical meridian focuses and the beam is a horizontal line. After that it simply spreads.

This shape is called Sturm’s conoid, after the nineteenth-century French mathematician Charles Sturm, who analysed it. Notice that each focal line lies at right angles to the meridian that forms it: the horizontal meridian focuses to a vertical line, and the vertical meridian to a horizontal one.

#The circle of least confusion

Between the two lines there is one place where the beam is round: the circle of least confusion. Of all the cross-sections it is the one whose widest dimension is smallest, and an astigmatic eye generally sees best when this circle falls on its retina. Notice where it is. The focal lines are at 33.3 cm and 100 cm, but the circle is not halfway between them in distance, at 66.7 cm. It is at 50 cm: halfway between them in vergence.

To see why, follow the width of the beam in each meridian. A ray leaving the edge of the lens, a distance $a$ from the centre, with vergence $V$, is $a(1-sV)$ from the axis after travelling a distance $s$. The beam is round where the widths in the two meridians are equal and opposite, one meridian having already crossed through its focus:

So the circle lies where the vergence would be the average of the two meridians, which is the spherical equivalent. For +1.00 / +2.00 × 090 that is +2.00 D, or 50 cm. Now change the cylinder of a lens whose spherical equivalent is held at +3.00 D:

The distance between the two lines is the interval of Sturm. It grows with the cylinder, but the circle of least confusion doesn’t move. That is why the spherical equivalent is the best single sphere for an astigmatic eye: it puts the circle of least confusion on the retina.

#Astigmatic eyes

An eye is astigmatic when its cornea, or less often its crystalline lens, curves more steeply in one meridian than in the other. Light from a point then forms two focal lines inside the eye, and the retina can catch at most one of them. Each principal meridian has its own refractive error, and the eye’s prescription is the spherocylinder that corrects both. We describe the astigmatism by where the two lines fall while the eye is relaxed. Try a few prescriptions:

When one line lies on the retina, the astigmatism is simple: simple myopic if the other is in front of the retina, simple hyperopic if it is behind. When both lie on the same side it is compound myopic or hyperopic, and when they straddle the retina it is mixed. In every case the correcting lens is the one whose principal meridians cancel the eye’s, which moves both lines onto the retina at once.

The orientation matters too. If the vertical meridian is the more powerful, as it is in most young eyes, the astigmatism is with the rule, corrected by a minus cylinder with its axis within 30° of 180. If the horizontal meridian is the more powerful, it is against the rule, corrected by a minus cylinder near 90, and anything in between is oblique. Over a lifetime, astigmatism tends to drift from with-the-rule toward against-the-rule.

#Finding the axis

How can we find the axis of an eye’s astigmatism just by asking the patient? One way is a chart of lines radiating like the hours on a clock, the clock dial or fan chart. The blur patch of an astigmatic eye is not round but stretched, so lines that run along the stretch stay fairly sharp, while lines across it smear out. Here is the dial as seen by an eye that needs −0.50 / −2.00 × 180. Turn the axis of its astigmatism:

For this to work, both focal lines must lie in front of the retina. Then the clearest lines are always those parallel to the focal line nearer the retina, and the patient has no reason to accommodate, which would only push both lines further forward. Clinicians make sure of it by fogging: adding enough plus to make the eye myopic in both meridians. This eye is fogged already, with −0.50 D of error in one meridian and −2.50 D in the other.

The clearest line lies at right angles to the axis of the correcting minus cylinder, and the numbers on the clock give a shortcut, the rule of 30: multiply the smaller hour number of the clearest line by 30. If the 12–6 line is clearest, the axis is 6 × 30 = 180; if it is the 1–7 line, the axis is 030. The clinician then adds minus cylinder at that axis until every line looks equally clear, and finally takes away the fog with the sphere.

#Refract a patient

Here is a patient whose prescription is hidden. They are myopic enough to be fogged already. Find their prescription:

1. Find the clearest line on the dial and use the rule of 30 to set the trial axis.
2. Add minus cylinder until every line looks the same.
3. Add minus sphere until the letters are as sharp as they will go, and no further.

If the cylinder goes in at the wrong axis, the lines never quite even out: two cylinders at different axes combine into a new spherocylinder, and the remaining astigmatism just turns to a new axis. The other classic mistake is too much minus sphere. The patient can accommodate through it and the chart still looks clear, which is why a refraction always ends on the most plus, or least minus, that gives the best vision.

#Recap

• A cylinder focuses parallel light to a line parallel to its axis. Its power in meridian θ is $C{\sin }^2(\theta -\alpha )$.
• Axes are measured anticlockwise from the horizontal as the examiner faces the patient, from 1 to 180, and written with three digits.
• A spherocylinder has two principal meridians, of power $S$ and $S+C$. To transpose: make $S+C$ the new sphere, change the sign of $C$ and turn the axis through 90°.
• Sturm’s conoid: two focal lines, each at right angles to the meridian that forms it, and between them the circle of least confusion, at the spherical equivalent $S+C/2$.
• Astigmatism is simple, compound or mixed; with the rule (minus cylinder axis near 180), against the rule (near 90) or oblique.
• On a fogged eye, the clearest line on the clock dial gives the minus cylinder axis: its smaller hour number × 30.

#Test yourself

#Further reading

• R. B. Rabbetts, Bennett and Rabbetts’ Clinical Visual Optics. Butterworth-Heinemann.
• M. P. Keating, Geometric, Physical, and Visual Optics. Butterworth-Heinemann.
• W. F. Harris, Dioptric power: its nature and its representation in three- and four-dimensional space. Optometry and Vision Science 74 (1997) 349–366.
• L. N. Thibos, W. Wheeler and D. Horner, Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error. Optometry and Vision Science 74 (1997) 367–375.