A simple interpretation of Zernike Polynomials
Few images in wavefront optics has been as common as Zernike Polynomials, yet it is a subject that has been obscured with trepidation and confusion for a long time for students who have their interest in the subject. Harry S Truman had once famously said, ‘ if you cannot convince them, confuse them’. In this write up, I will however try my best to shed some light on Zernike Polynomials and how it can be read from an Ophthalmological perspective without anyway trying to confuse you :-)
The Zernike Polynomials are depicted by the symbol :
The symbol Z has a numerator/superior(m) and a denominator/inferior (n). In strict physics term, the definition of the ‘m’ would stand for a continuous function that repeats self every 2π radians, while the ‘n’ would stand for a radial function that is proportional to the order of polynomials.
However to us, who are connected to the science of eye, this definition would not make us comfortable, and therefore one of the reasons why the Zernike polynomials have been that least understood subject in our domain.
How do we interpret Zernike Polynomials in a more simplified way that connects with us in ophthalmology?
The Zernike function denomination in (fig 1A) (and for the simplicity to type in my computer Z m/n ) can be interpreted in ophthalmology thus–
m- number of affected (principal) meridians of the cornea. n- the order of the aberration ( the order of n is given in Fig 2 at it's left ) or if you look in Fig 1 or in Fig 2 from top to bottom of the rows, then the top most single aberration, called piston aberration, is your first row or n=0, as you walk down you go through n1(second row ),n2 (third row ) and so on.
Before we move on to further explain, I must clear a few basic concept.
First, in the first two rows or orders ( n= 0 and n=1) the aberrations are also called constant aberrations (fig 1). The n=0 or zero order is also called piston aberration. This aberration can be equated with the piston of an engine (fig 3) wherein the piston motion is vertically moving up and down. As the piston movement is vertically upwards, and comes more closer to our eyes, the object that is the piston looks bigger. The more the piston moves downward and away from our eyes, the object becomes smaller. In Ophthalmology, this is not a true aberration, but only an image size variation. Since only the size of the object image is varied, we call it as a quantitative variation and not a qualitative variation. This aberration therefore is not important clinically.
You will also notice that when the sun and the moon are together visible in the pale sky, both of them seem to be of equal size. However the sun is a few hundred times bigger than the moon, but we see it of the same size because the moon is also closer to us than the sun.
Now let us try to understand the Zernike polynomial attached to this aberration as seen in Fig 1. It is ( m=0 and n=0) or
The superior/numerator ‘m’ equals to zero because none of the principal meridians in the 360 degree meridians are affected, while the inferior/denominator ‘n’ or the radial order is 0 as the start of the polynomial functions are 0 ( row one or n=0)
Second, the next order of aberration (n=1 ) is the aberration of tilt. If you see a tall pole standing straight from the ground, from an ‘off axis’ or from an angular point, then at certian points the image may look a little tilted. Often a square or a rectangle when viewed from an angle or an off axis position may look like a rhomboid shape.This is referred to as the first order of aberration of vertical or horizontal tilt. This is not a true aberration as the wavefront becomes aberration free with a change in position.
In the picture here in Fig 4, we know that all the columns are truly vertical, but as we see the picture from an angle, to us it seems as tilted. This is again not a qualitative aberration, but a correction can be made by a mere change of position of the observer, or in this case, the picture can be taken more from a position that is straight in line before the object. This aberration is of little clinical significance to us as this need not be corrected with any intervention, either with glasses or any other means. The aberration is denoted by m=-1, and n=1. The m=-1, or
here the m or superior/numerator is 1 because one of the principal meridians in the eye are affected because of the tilt involved in viewing the object from an angle, while n=1 because that is the order in the Zernike polynomial function depicted in Figure 1.
Third, aberrations with a (-) and (+) sign:
One important thing to state here is how aberrations in Zernike polynomials can be denoted with (-) or a (+) sign. In figure 5, you can see all aberrations to the left (vertical) are denoted in (-) while in the right (horizontal) are denoted by (+). That explains why all aberrations from the n=1 to beyond are denoted by either a plus or a minus sign. Thus in first order (n=1), prismatic tilt have both vertical and horizontal forms, if the tilt is more vertically oriented it is denoted by a minus sign, while for the horizontal it is the opposite.
The next order in Zernike Polynomials is the lower order aberrations that are clinically significant. Most of us are very familiar with these aberrations of defocus ( myopia or hypermetropia) and astigmatism. Here now, we are discussing about n=2 or the second order of aberrations. In Figure 1 the second order or n=2 starts with vertical (-) astigmatism also denoted by Z-2/2 . The numerator 2 denotes that both the principal meridians are affected, while the denominator 2 denotes the order of the aberration in a radial function. To explain more why the numerator is denoted by '2', think of how many principal meridians are affected by regular astigmatism. Its two, is it not ? The steeper and the flatter meridians of the cornea.
Hence m=(-/+)2, or
Next in the same radial order or n=2, you can see in Figure 1 the aberration of defocus with Zernike polynomial of Z0/2. The m or numerator is zero because all meridians are homogenously affected, and the denominator or n is 2 as this is again an aberration of the second order. As we move towards the right in the same order (n=2), we get horizontal astigmatism with the function of Z2/2. I am sure by now you will yourself be able to describe the logic of this function.
All other orders beyond the n=2 or second order are Higher Order Aberrations. There are infinite number of aberrations of higher order like a Kaleidoscope, however for clinical significance in Ophthalmology, we only describe the third and fourth order of aberrations. In the third order or n=3, we have vertical trefoil, vertical coma, horizontal coma and horizontal trefoil. These are described in Zernike polynomials as Z-3/3 ( vertical trefoil), Z-1/3 (vertical coma), Z1/3 (horizontal coma), Z3/3 ( horizontal trefoil). The denotation of the Zernike polynomials all follow the same logic as described in the previous orders.
For example, in Fig 1, the vertical trefoil is denoted as Z-3/3 because the denominator or 'n' is the function of the third order while the numerator or 'm' is minus (vertical) and the three(3) because they affect three principal meridians affected with trefoil (see below picture)
The horizontal trefoil in the far right of the third order(n=3) will be also similar, but positive.
Remember the word 'trefoil' denoting ' three ' & 'foil', that is the aberration is spread out in three directions affecting three meridians much like the symbol of the Mercedes Benz company. Another example could be the symbol of the three leaves from the Gothic architecture signifying the Generator, operator and destroyer, the trinity of God. For Hindus, the trefoil 'bilipatra' is quite common for offering to Lord Shiva. If a point source of light is seen through a trefoil aberration, we will see the light in three direction, albeit, smeared.
The Coma, both vertical and horizontal, comes from the word comet, because of its similarity with the later having a head and a tail. If a light is coming from an off axis point, that is the light coming from infinity is not parrallel to the optical axis of the lens, all the light will not come to focus at one point, but rather would give rise to a comet like aberration. This is because rays that pass closer to the optical axis will be of relatively sharply focussed, but as the rays of light passes through the periphery of the lens, the light will be less focussed on the retina. The resultant effect of this is like a coma like aberration.For eyes with more amount of angle alpha will also suffer from such aberration.
Beyond the third order of aberration, we only have to look at the spherical aberration as visually significant ( in the middle of the fifth line in fig 1 when you count from above). The spherical aberration is of the fourth order, and by far the most visually significant of all the HOA. I am sure by now you are able to justify the Zernike polynomial describing this aberration in Fig 1. The spherical aberratin of of the fourth order aberration and hence the denominator is n=4 while since all the meridians of the periphery are equally affected, the numerator or m stands zero.
Several studies have pointed out to the deleterious effects of three HOA on human vision - spherical aberration, coma and trefoil. Therefore the focus of this write up was to explain the Zernike Polynomials related to the aberrations more affecting the visual acuity or contrast sensitivity.
All the best !
A video explanation to this write up is available at : https://www.youtube.com/watch?v=NmdmWJ6lXq0