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There are many fields, including astronomy, lasers, fiber optics, optometry and others where it is useful to be able to represent the surface of a wavefront or even the surface of an optical component (but more importantly the wavefront changes due to that component). The Zernike polynomials provide a complete orthogonal set of basis functions for this purpose as long as you are willing to confine the footprint of the wavefront or surface to a circular region.
Zernike polynomial representation of a wavefront is done in the form of an infinite series. Fortunately, the significance of the terms drops off nicely as you go to higher orders just as you would expect for any useful infinite series.
Equation 1
Where the coefficients are:
Equation 2
Or
Equation 3
Which (after careful analysis) always turns out to be:
Equation 4
Or
Equation 5
This works for ALL terms including l=0 terms.
Equation 6
The following table lists the first 35 Zernike polynomial terms in the order that I use for the Zernike representation of a wavefront. These polynomials were first introduced by F. Zernike in 1934[*]This treatment is based on the description by Born and Wolf [†] where you can find a derivation based on the orthogonality and invariance requirements. For our purposes the Zernike polynomials are a set of orthogonal polynomials on a unit circle which work very well to describe a surface representing the constant phase surface of an optical wavefront.
Each unique polynomial,
which is a function of polar coordinates on the unit circle, is specified by two
indices, 
, and 
and can be broken
into a radial function and a sinusoidal part by polar angle.
The index, 
, can be any positive integer or zero, and 
can be any integer from 
to 
We will use the real polynomials, which are constructed from the complex polynomials by:
From this you can see
that the sign of the second index determines which sinusoidal function
describes our wavefront surface as we sweep the angle, 
. Our radial function is the same for either
sign and is given by:
| Zernike Terms for Wavefront Representation | |||
| # | Form of the Polynomial | (n,m) | Name |
| 0 | 1 | (0,0) | Piston |
| 1 |
|
(1,1) | Tilt X |
| 2 |
|
(1,-1) | Tilt Y |
| 3 |
|
(2,2) | Astigmatism X |
| 4 |
|
(2,0) | Power or Focus |
| 5 |
|
(2,-2) | Astigmatism Y |
| 6 |
|
(3,3) | Trefoil X |
| 7 |
|
(3,1) | Coma X |
| 8 |
|
(3,-1) | Coma Y |
| 9 |
|
(3,-3) | Trefoil Y |
| 10 |
|
(4,4) | Tetrafoil X |
| 11 |
|
(4,2) | Secondary Astigmatism X |
| 12 |
|
(4,0) | Primary Spherical |
| 13 |
|
(4,-2) | Secondary Astigmatism Y |
| 14 |
|
(4,-4) | Tetrafoil Y |
| 15 |
|
(5,5) | Pentafoil X |
| 16 |
|
(5,3) | Secondary Trefoil X |
| 17 |
|
(5,1) | Secondary Coma X |
| 18 |
|
(5,-1) | Secondary Coma Y |
| 19 |
|
(5,-3) | Secondary Trefoil Y |
| 20 |
|
(5,-5) | Pentafoil Y |
| 21 |
|
(6,6) | Hexafoil X |
| 22 |
|
(6,4) | Secondary Tetrafoil X |
| 23 |
|
(6,2) | Tertiary Astigmatism X |
| 24 |
|
(6,0) | Secondary Spherical |
| 25 |
|
(6,-2) | Tertiary Astigmatism Y |
| 26 |
|
(6,-4) | Secondary Trefoil Y |
| 27 |
|
(6,-6) | Hexafoil Y |
| 28 |
|
(7,7) | Heptafoil X |
| 29 |
|
(7,5) | Secondary Pentafoil X |
| 30 |
|
(7,3) | Tertiary Trefoil X |
| 31 |
|
(7,1) | Tertiary Coma X |
| 32 |
|
(7,-1) | Tertiary Coma Y |
| 33 |
|
(7,-3) | Tertiary Trefoil Y |
| 34 |
|
(7,-5) | Secondary Pentafoil Y |
| 35 |
|
(7,-7) | Heptafoil Y |
Frits Zernike (1888-1966) Frederik (Frits) Zernike (or Zernicke) was born July 16, 1888 in Amsterdam. He earned a doctorate in chemistry from the Universty of Amsterdam in 1915 and became a lecturer in physics at the University of Groningen becoming a full professor by 1920.
Zernike's research for that first decade was focused on statistical mechanics. He worked on optics in the 1930's where one of his contributions was "phase contrast microscopy" which exploits the phase differences produced by transmission through different materials or tissues to provide contrast between tissues even in living organisms without the need to use dyes or stains.
Zernike was awarded the Nobel prize in 1953. He obtained a patent for his invention of phase contrast microscopy in 1936.
[*] F. Zernike, Physica, 1 (1934), 689.
[†] M. Born, E. Wolf, Principles of Optics -6th ed., Cambridge University Press, 1997.
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