Take a look at figure 1. What is it? A work of art? Maybe. To paraphrase Picasso: ‘Art is the lie which reveals the truth.’ More prosaically it is simply a three-dimensional diagram depicting ray paths in astigmatic refraction.
Understanding the nature of astigmatic refraction can be a challenge to those new to ophthalmic optics. My motivation to produce a model was to help demonstrate the salient features and to share the construction details should anyone wish to replicate it.
Two-dimensional limitation
Most of us involved with ophthalmic optics will be familiar with two dimensional diagrams similar to the one shown in figure 2. Here we see the location of the principal ‘landmarks’ of an astigmatic beam: the two focal lines and the circle of least confusion. In optical terms the region between the two focal lines is known as the interval of Sturm. In geometrical terms the space bounded by the rays within the interval of Sturm is known as the conoid of Sturm, named after the Swiss-born mathematician Jacques-Charles-François Sturm (1803-55).
However, the two-dimensional diagram is confusing with respect to the interval of Sturm. What is happening here? Are half the rays converging and the other half diverging? Actually, no, all the rays outside of the principal meridians are both converging and diverging at the same time. How is that possible? It depends how you look at it, and the best way to look is in three dimensions.
Figure 2: The red and green rays are in mutually perpendicular meridians. The blue lines denote the locations (from left to right) of the first focal line, the circle of least confusion and the second focal line
The interval of Sturm
The basic framework of ‘A Perfect Sturm’ is, I think, self-explanatory from figure 1, consisting of two square acrylic end-plates held by four legs secured with nuts and washers. The red tape on one leg is a temporary aid to recognise the orientation of the model.
The end-plates used were 16x16cm sheets of 4.5mm acrylic. The legs were 23cm lengths of 8mm threaded steel rod. The ‘rays’ were comprised of stretched black elastic thread approximately 0.8mm thick.
Both end-plates are identical in design. The ray hole locations are based on the geometry of a 15cm circle divided into 15 degree steps then scaled 25% vertically (figure 3, upper). The end-plates are mounted perpendicular to each other (figure 3, lower).
Figure 3: Locating the ray holes (upper image). The end-plates are disposed perpendicularly (lower image) – upper plate ray holes shown in black, lower in red
The ray holes are drilled just large enough to allow free passage of the elastic thread but too small to pass when knotted. It is wise to number the holes sequentially with non-permanent marker, starting with the 12 o’clock position on the upper end-plate and the 6 o’clock position on the lower plate proceeding clockwise in both cases (figure 4, upper). Corresponding numbered ray holes on both end-plates are threaded through with a single piece of elastic thread, stretched taught and knotted at both ends, any excess being subsequently trimmed off. Figure 4 (lower) shows the completed sequence of threading.
Figure 4: Arrangement of ray hole connections between upper and lower end-plates (upper image). Completed connections between end-plates (lower image)
A card circle denoting the circle of least confusion can be inserted within the centre of the ray thread bundle where it will nestle without further support. Its diameter needs to be 3/8 of the initial diameter chosen to generate the ray holes (which in this case was 15cm, hence the card’s diameter was 5.625cm).
A further, optional, refinement of this design was to include the chief (undiverted) ray through the centre of the system, this was executed with white elastic thread instead of black.
Figure 5: Two end-on views of ‘A Perfect Sturm’ showing ray concentration at the focal lines
This model presents an idealised (hence ‘perfect’) representation of Sturm’s conoid insofar as the two focal lines are equal due to the symmetry of the construction. It can be seen from figure 2 that theoretically these are unequal with the second focal line invariably being the longer. However, when we consider moderate astigmatism within the eye, their difference is small enough to be disregarded.
Figure 6: Horizontal: 90 degree rotation about the vertical axis
Vertical: 180 degree rotation about a horizontal axis
An interesting aspect of this model is that all the individual ‘rays’ (aside from the chief ray) have the same length. This implies, since the end-plates are parallel, that they all have the same angle of deviation from the vertical (approximately 24 degrees).
Another interesting aspect of this model’s geometry is its isomorphic property. The transformation effected by a 90 degree rotation about the vertical axis results in identical geometry to a 180 degree rotation about a horizontal axis (figures 5 and 6, note the location of the red tape).
Henry Burek is an optometrist resident in South Yorkshire and an examiner for the College Of Optometrists.