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 :

Fig 1A

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.

Next we will look into other higher order aberrations as defined by zernike polynomials.

Click on the video below to go through the entire concept of how to interpret zernike polynomials for application in ophthalmology

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