In this article, I will explain astigmatism and how it is corrected with glasses. To explain in a lucid way, we will talk in terms of corneal astigmatism or cylinder, and its correction with prescription glasses.
Imagine you visit your doctor and he/she finds out that you have astigmatism. This is subjective refraction, the sum total of the corneal and lenticular refractions (total eye refraction). It shows that you have astigmatism, that is in one meridian you have more (or less) power, while the meridian which is 90 degree away has just the right power. To keep matters simple, let us keep the human crystalline lens (HCL) away, and assume that the astigmatism is solely due to cornea. So with one meridian of cornea more (or less power) than average, while the other meridian which is 90 degree away having a normal power, will make the refraction of your cornea a cylinder in shape (Fig1)
What does a cylinder look like ? A cylinder, in optics, and as depicted in figure 1 has one side curved, and the other side flat. The side which is flat (y axis in fig1) has no power and is called axis. In the fig1 the axis or the meridian of no power is at 90 degree. The side which has power is called power meridian, that is 180 degree in the picture (x axis in fig1). Here the horizontal side has power and the vertical side is the Axis, or a place of no power. The difference between the Principal Meridian (power meridian) and the axis (no power) is astigmatism.
To be more scientific, and talking in corneal terms, we understand the average power of the cornea is around 43.50 dioptres. What this means in the context of the above described figure is that one meridian will have just the right power (normal power), while the other meridian which is 90 degree away will have either more or less power than 43.50 dioptres. For example, in the context of Fig 1, the patient's cornea will have 43.50 dioptres on the y axis (90 degree), but has more power (say 46.50 dioptres) at the x axis (180 degrees). The rays of light that pass through the 90 degree will therefore fall on the retina, while the rays of light that pass through 180 degree will fall before the retina as because the cornea is too steep at the horizontal axis.
How to correct ? A glass given to you would have a cylinder shape, but would be placed in the opposite meridian (Fig2). So here the glass cylinder will be aligned horizontally, that is the flat axis of the glass will be aligned on the x axis or 180 deg meridian ( Fig3 ). Thus the power meridian aligned to 90 deg of the prescription cylinder ( depicted in yellow colour in Fig2 and Fig3) , will negate the corneal Power meridian at 180 deg ( depicted in grey cylinder in Fig1). The result, you have is a spherical refraction depicted in green (Fig3).
Going by our previous example in corneal terms, such a patient would be given a prescription/subjective refraction written like this : +3.50@180( power meridian of 3.5 D will be then aligned to 90 degree, that is to the flat meridian of cornea) to negate the +3.50 diotres of more power the patient has in cornea at 180 degree ( 46.50@180 and 43.50@90 on cornea).
Now let us consider, you have both astigmatism as well as myopia (or hyperopia) found by your doctor. How will your cornea look like ? It would look like a doughnut ( see Fig 4 ).
Unlike the cylinder, a doughnut would not have any flat surface. All surfaces will be curved. However, there will be one surface more curved than the other. In Fig4 you can see the tube representing the cornea has one axis more curved than the other axis. Or in other words, in the Fig4, the y-axis or 90 degree is significantly more curved than the x-axis (180 degree).
Such cornea can be described like a rugby ball also (Fig5). Like the doughnut example, here again both meridians are curved, but one meridian is more curved than the other. Since both sides have power, they will be called power meridians, or Principal Meridians.
To state in terms of corneal example, both axis of cornea (90 degree and 180 degree) have power which is either more or less than average power of cornea. In our last example, remember the patient had more power than average cornea (46.50 dioptres) at 180 degree. But in the other meridian that was 90 degree away, the patient had normal power ( 43.50 dioptre). In this example however, the patient will have more power or less power than 43.50 dioptre in both the vertical and horizontal meridians. Say in this case the patient has 44.50 dioptre @ 90 degree. So the patient has 46.50@180 and 44.50@90, and therefore in both the meridians the patient has more power than average and normal cornea.
Astigmatism here is the difference between the two principal meridians. That is 46.50 - 44.50 = 2 dioptres. Thus we can represent this as +2.0 at 180 degree. We can say here that the 180 degree is steeper than an average cornea of 43.50 dioptres by +3 dioptres, and 90 degree is steeper by +1 dioptre. Thus both meridians are steeper than normal conea, and rays of light passing through both the meridians of 90 and 180 degree are falling before the retina. The patient's cornea is having a compound myopic astigmatism, that is both meridians have stronger power on the cornea than average cornea. To correct this, we need to give a equally rughby shaped glass, but have to place the same power in opposite meridians to negate the difference in corneal power. Thus since the patient has +3 at 180 degee and +1 at 90 degree on cornea, we will have to negate it with a -3@180 degree and -1 at 90 degree on the glass,
Thus, the concept of correction of astigmatism with myopia (or hyperopia) is the same as that of the patient with astigmatism only (in the first example). The patient with a difference between two principal meridians ( astigmatism ) will have to be given a similar difference of two meridian power in the glass, albeit, on the opposite meridians/directions. Such lenses, whether glass or an IOL ( if a cataract patient ) are also called sphero- cylindrical lenses. The TORIC IOLs of different surgical companies implanted intraoperatively, is a sphero-cylindrical lens where you have a difference of power in the two opposite meridian.
Since we started by describing that we will talk about astigmatism in corneal terms, let us assume that the patient's corneal reading ( Keratometry ) is +2 at 90 degree. That is the patient' 90 degree (y axis) has more power than the 180 degree, which is often labelled as with-the-rule astigmatism. How will we correct this corneal astigmatism with glass?
Such a patient will have a subjective refraction typically like this, that is the optometrist will check his eye and see that he may be accepting a glass power of -1 -2 @90
-1-2@90, whereby -1 denotes myopia the patient is having, the -2 denotes the difference in principal meridians (astigmatism), and the axis 90 degree is where you are placing the spherical power in glass as -1 to correct for the difference in power of the two principal meridians. In the world of physicians and optometrists, this is also referred to as an optical cross.
If you do a transposition then you get the second principal meridian. The concept of transposition is described in Eye & Astigmatism series in this site Video section.
So, when you transpose -1-2@90 would be -3+2@180. To do transposition, add the spherical and the cylinder, then change the sign before the cylinder , keep the magnitude of astigmatism unchanged, and change the axis by 90 deg.
Therefore the two spherical powers are -1 and -3 diopters in the glass. The patient's glass will have -1 at 90 degree while -3 will be put at 180 degree. Thus the difference of 2 dioptres in glass power on 180 degree will negate the +2 of astigmatism at the corneal plane.