Zernike polynomials

Rewriting the ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers:

${\displaystyle R_{n}^{m}(\rho )=\sum _{k=0}^{\tfrac {n-m}{2}}(-1)^{k}{\binom {n-k}{k}}{\binom {n-2k}{{\tfrac {n-m}{2}}-k}}\rho ^{n-2k}}$.

A notation as terminating Gaussian hypergeometric functions is useful to reveal recurrences, to demonstrate that they are special cases of Jacobi polynomials, to write down the differential equations, etc.:

${\displaystyle {\begin{aligned}R_{n}^{m}(\rho )&={\binom {n}{\tfrac {n+m}{2}}}\rho ^{n}\ {}_{2}F_{1}\left(-{\tfrac {n+m}{2}},-{\tfrac {n-m}{2}};-n;\rho ^{-2}\right)\\&=(-1)^{\tfrac {n-m}{2}}{\binom {\tfrac {n+m}{2}}{m}}\rho ^{m}\ {}_{2}F_{1}\left(1+{\tfrac {n+m}{2}},-{\tfrac {n-m}{2}};1+m;\rho ^{2}\right)\end{aligned}}}$

for n − m even.

The factor ${\displaystyle \rho ^{n-2k}}$ in the radial polynomial ${\displaystyle R_{n}^{m}(\rho )}$ may be expanded in a Bernstein basis of ${\displaystyle b_{s,n/2}(\rho ^{2})}$ for even ${\displaystyle n}$ or ${\displaystyle \rho }$ times a function of ${\displaystyle b_{s,(n-1)/2}(\rho ^{2})}$ for odd ${\displaystyle n}$ in the range ${\displaystyle \lfloor n/2\rfloor -k\leq s\leq \lfloor n/2\rfloor }$. The radial polynomial may therefore be expressed by a finite number of Bernstein Polynomials with rational coefficients:

${\displaystyle R_{n}^{m}(\rho )={\frac {1}{\binom {\lfloor n/2\rfloor }{\lfloor m/2\rfloor }}}\rho ^{n\mod 2}\sum _{s=\lfloor m/2\rfloor }^{\lfloor n/2\rfloor }(-1)^{\lfloor n/2\rfloor -s}{\binom {s}{\lfloor m/2\rfloor }}{\binom {(n+m)/2}{s+\lceil m/2\rceil }}b_{s,\lfloor n/2\rfloor }(\rho ^{2}).}$

Applications often involve linear algebra, where an integral over a product of Zernike polynomials and some other factor builds a matrix elements. To enumerate the rows and columns of these matrices by a single index, a conventional mapping of the two indices n and l to a single index j has been introduced by Noll.[3] The table of this association ${\displaystyle Z_{n}^{l}\rightarrow Z_{j}}$ starts as follows (sequence A176988 in the OEIS). ${\displaystyle j={\frac {n(n+1)}{2}}+|l|+\left\{{\begin{array}{ll}0,&l>0\land n\equiv \{0,1\}{\pmod {4}};\\0,&l<0\land n\equiv \{2,3\}{\pmod {4}};\\1,&l\geq 0\land n\equiv \{2,3\}{\pmod {4}};\\1,&l\leq 0\land n\equiv \{0,1\}{\pmod {4}}.\end{array}}\right.}$

The rule is the following.

• The even Zernike polynomials Z (with even azimuthal parts ${\displaystyle \cos(m\varphi )}$, where ${\displaystyle m=l}$ as ${\displaystyle l}$ is a positive number) obtain even indices j.
• The odd Z obtains (with odd azimuthal parts ${\displaystyle \sin(m\varphi )}$, where ${\displaystyle m=\left\vert l\right\vert }$ as ${\displaystyle l}$ is a negative number) odd indices j.
• Within a given n, a lower ${\displaystyle \left\vert l\right\vert }$ results in a lower j.

OSA [4] and ANSI single-index Zernike polynomials using:

${\displaystyle j={\frac {n(n+2)+l}{2}}}$

The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g., photolithography.[5][6]

${\displaystyle j=\left(1+{\frac {n+|l|}{2}}\right)^{2}-2|l|+{\frac {1-\operatorname {sgn} l}{2}}}$

where ${\displaystyle \operatorname {sgn} l}$ is the sign or signum function. The first 20 fringe numbers are listed below.

James C. Wyant uses the "Fringe" indexing scheme except it starts at 0 instead of 1 (subtract 1).[7] This method is commonly used including interferogram analysis software in Zygo interferometers and the open source software DFTFringe.

The orthogonality in the radial part reads[8]

${\displaystyle \int _{0}^{1}{\sqrt {2n+2}}R_{n}^{m}(\rho )\,{\sqrt {2n'+2}}R_{n'}^{m}(\rho )\,\rho d\rho =\delta _{n,n'}}$

or

${\displaystyle {\underset {0}{\overset {1}{\mathop {\int } }}}\,R_{n}^{m}(\rho )R_{{n}'}^{m}(\rho )\rho d\rho ={\frac {{\delta }_{n,{n}'}}{2n+2}}.}$

Orthogonality in the angular part is represented by the elementary

${\displaystyle \int _{0}^{2\pi }\cos(m\varphi )\cos(m'\varphi )\,d\varphi =\epsilon _{m}\pi \delta _{m,m'},}$

${\displaystyle \int _{0}^{2\pi }\sin(m\varphi )\sin(m'\varphi )\,d\varphi =\pi \delta _{m,m'};\quad m\neq 0,}$

${\displaystyle \int _{0}^{2\pi }\cos(m\varphi )\sin(m'\varphi )\,d\varphi =0,}$

where ${\displaystyle \epsilon _{m}}$ (sometimes called the Neumann factor because it frequently appears in conjunction with Bessel functions) is defined as 2 if ${\displaystyle m=0}$ and 1 if ${\displaystyle m\neq 0}$. The product of the angular and radial parts establishes the orthogonality of the Zernike functions with respect to both indices if integrated over the unit disk,

${\displaystyle \int Z_{n}^{l}(\rho ,\varphi )Z_{n'}^{l'}(\rho ,\varphi )\,d^{2}r={\frac {\epsilon _{l}\pi }{2n+2}}\delta _{n,n'}\delta _{l,l'},}$

where ${\displaystyle d^{2}r=\rho \,d\rho \,d\varphi }$ is the Jacobian of the circular coordinate system, and where ${\displaystyle n-l}$ and ${\displaystyle n'-l'}$ are both even.

Any sufficiently smooth real-valued phase field over the unit disk ${\displaystyle G(\rho ,\varphi )}$ can be represented in terms of its Zernike coefficients (odd and even), just as periodic functions find an orthogonal representation with the Fourier series. We have

${\displaystyle G(\rho ,\varphi )=\sum _{m,n}\left[a_{m,n}Z_{n}^{m}(\rho ,\varphi )+b_{m,n}Z_{n}^{-m}(\rho ,\varphi )\right],}$

where the coefficients can be calculated using inner products. On the space of ${\displaystyle L^{2}}$ functions on the unit disk, there is an inner product defined by

${\displaystyle \langle F,G\rangle$ :=\int F(\rho ,\varphi )G(\rho ,\varphi )\rho d\rho d\varphi .}

The Zernike coefficients can then be expressed as follows:

${\displaystyle {\begin{aligned}a_{m,n}&={\frac {2n+2}{\epsilon _{m}\pi }}\left\langle G(\rho ,\varphi ),Z_{n}^{m}(\rho ,\varphi )\right\rangle ,\\b_{m,n}&={\frac {2n+2}{\epsilon _{m}\pi }}\left\langle G(\rho ,\varphi ),Z_{n}^{-m}(\rho ,\varphi )\right\rangle .\end{aligned}}}$

Alternatively, one can use the known values of phase function G on the circular grid to form a system of equations. The phase function is retrieved by the unknown-coefficient weighted product with (known values) of Zernike polynomial across the unit grid. Hence, coefficients can also be found by solving a linear system, for instance by matrix inversion. Fast algorithms to calculate the forward and inverse Zernike transform use symmetry properties of trigonometric functions, separability of radial and azimuthal parts of Zernike polynomials, and their rotational symmetries.

The reflections of trigonometric functions result that the parity with respect to reflection along the x axis is

${\displaystyle Z_{n}^{l}(\rho ,\varphi )=Z_{n}^{l}(\rho ,-\varphi )}$ for l ≥ 0,

${\displaystyle Z_{n}^{l}(\rho ,\varphi )=-Z_{n}^{l}(\rho ,-\varphi )}$ for l < 0.

The π shifts of trigonometric functions result that the parity with respect to point reflection at the center of coordinates is

${\displaystyle Z_{n}^{l}(\rho ,\varphi )=(-1)^{l}Z_{n}^{l}(\rho ,\varphi +\pi ),}$

where ${\displaystyle (-1)^{l}}$ could as well be written ${\displaystyle (-1)^{n}}$ because ${\displaystyle n-l}$ as even numbers are only cases to get non-vanishing Zernike polynomials. (If n is even then l is also even. If n is odd, then l is also odd.) This property is sometimes used to categorize Zernike polynomials into even and odd polynomials in terms of their angular dependence. (it is also possible to add another category with l = 0 since it has a special property of no angular dependence.)

• Angularly even Zernike polynomials: Zernike polynomials with even l so that ${\displaystyle Z_{n}^{l}(\rho ,\varphi )=Z_{n}^{l}(\rho ,\varphi +\pi ).}$
• Angularly odd Zernike polynomials: Zernike polynomials with odd l so that ${\displaystyle Z_{n}^{l}(\rho ,\varphi )=-Z_{n}^{l}(\rho ,\varphi +\pi ).}$

The radial polynomials are also either even or odd, depending on order n or m:

${\displaystyle R_{n}^{m}(\rho )=(-1)^{n}R_{n}^{m}(-\rho )=(-1)^{m}R_{n}^{m}(-\rho ).}$

These equalities are easily seen since ${\displaystyle R_{n}^{m}(\rho )}$ with an odd (even) m contains only odd (even) powers to ρ (see examples of ${\displaystyle R_{n}^{m}(\rho )}$ below).

The periodicity of the trigonometric functions results in invariance if rotated by multiples of ${\displaystyle 2\pi /l}$ radian around the center:

${\displaystyle Z_{n}^{l}\left(\rho ,\varphi +{\tfrac {2\pi k}{l}}\right)=Z_{n}^{l}(\rho ,\varphi ),\qquad k=0,\pm 1,\pm 2,\cdots .}$

The Zernike polynomials satisfy the following recurrence relation which depends neither on the degree nor on the azimuthal order of the radial polynomials:[9]

${\displaystyle {\begin{aligned}R_{n}^{m}(\rho )+R_{n-2}^{m}(\rho )=\rho \left[R_{n-1}^{\left|m-1\right|}(\rho )+R_{n-1}^{m+1}(\rho )\right]{\text{ .}}\end{aligned}}}$

From the definition of ${\displaystyle R_{n}^{m}}$ it can be seen that ${\displaystyle R_{m}^{m}(\rho )=\rho ^{m}}$ and ${\displaystyle R_{m+2}^{m}(\rho )=((m+2)\rho ^{2}-(m+1))\rho ^{m}}$. The following three-term recurrence relation[10] then allows to calculate all other ${\displaystyle R_{n}^{m}(\rho )}$:

${\displaystyle R_{n}^{m}(\rho )={\frac {2(n-1)(2n(n-2)\rho ^{2}-m^{2}-n(n-2))R_{n-2}^{m}(\rho )-n(n+m-2)(n-m-2)R_{n-4}^{m}(\rho )}{(n+m)(n-m)(n-2)}}{\text{ .}}}$

The above relation is especially useful since the derivative of ${\displaystyle R_{n}^{m}}$ can be calculated from two radial Zernike polynomials of adjacent degree:[10]

${\displaystyle {\frac {\operatorname {d} }{\operatorname {d} \!\rho }}R_{n}^{m}(\rho )={\frac {(2nm(\rho ^{2}-1)+(n-m)(m+n(2\rho ^{2}-1)))R_{n}^{m}(\rho )-(n+m)(n-m)R_{n-2}^{m}(\rho )}{2n\rho (\rho ^{2}-1)}}{\text{ .}}}$

The first few radial polynomials are:

${\displaystyle R_{0}^{0}(\rho )=1\,}$

${\displaystyle R_{1}^{1}(\rho )=\rho \,}$

${\displaystyle R_{2}^{0}(\rho )=2\rho ^{2}-1\,}$

${\displaystyle R_{2}^{2}(\rho )=\rho ^{2}\,}$

${\displaystyle R_{3}^{1}(\rho )=3\rho ^{3}-2\rho \,}$

${\displaystyle R_{3}^{3}(\rho )=\rho ^{3}\,}$

${\displaystyle R_{4}^{0}(\rho )=6\rho ^{4}-6\rho ^{2}+1\,}$

${\displaystyle R_{4}^{2}(\rho )=4\rho ^{4}-3\rho ^{2}\,}$

${\displaystyle R_{4}^{4}(\rho )=\rho ^{4}\,}$

${\displaystyle R_{5}^{1}(\rho )=10\rho ^{5}-12\rho ^{3}+3\rho \,}$

${\displaystyle R_{5}^{3}(\rho )=5\rho ^{5}-4\rho ^{3}\,}$

${\displaystyle R_{5}^{5}(\rho )=\rho ^{5}\,}$

${\displaystyle R_{6}^{0}(\rho )=20\rho ^{6}-30\rho ^{4}+12\rho ^{2}-1\,}$

${\displaystyle R_{6}^{2}(\rho )=15\rho ^{6}-20\rho ^{4}+6\rho ^{2}\,}$

${\displaystyle R_{6}^{4}(\rho )=6\rho ^{6}-5\rho ^{4}\,}$

${\displaystyle R_{6}^{6}(\rho )=\rho ^{6}.\,}$

The first few Zernike modes, at various indices, are shown below. They are normalized such that: ${\displaystyle \int _{0}^{2\pi }\int _{0}^{1}Z^{2}\cdot \rho \,d\rho \,d\phi =\pi }$, which is equivalent to ${\displaystyle \operatorname {Var} (Z)_{\text{unit circle}}=1}$.

${\displaystyle Z_{n}^{l}}$	OSA/ANSI index (${\displaystyle j}$)	Noll index (${\displaystyle j}$)	Wyant index (${\displaystyle j}$)	Fringe/UA index (${\displaystyle j}$)	Radial degree (${\displaystyle n}$)	Azimuthal degree (${\displaystyle l}$)	${\displaystyle Z_{j}}$	Classical name
${\displaystyle Z_{0}^{0}}$	00	01	00	01	0	00	${\displaystyle 1}$	Piston (see, Wigner semicircle distribution)
${\displaystyle Z_{1}^{-1}}$	01	03	02	03	1	−1	${\displaystyle 2\rho \sin \phi }$	Tilt (Y-Tilt, vertical tilt)
${\displaystyle Z_{1}^{1}}$	02	02	01	02	1	+1	${\displaystyle 2\rho \cos \phi }$	Tilt (X-Tilt, horizontal tilt)
${\displaystyle Z_{2}^{-2}}$	03	05	05	06	2	−2	${\displaystyle {\sqrt {6}}\rho ^{2}\sin 2\phi }$	Oblique astigmatism
${\displaystyle Z_{2}^{0}}$	04	04	03	04	2	00	${\displaystyle {\sqrt {3}}(2\rho ^{2}-1)}$	Defocus (longitudinal position)
${\displaystyle Z_{2}^{2}}$	05	06	04	05	2	+2	${\displaystyle {\sqrt {6}}\rho ^{2}\cos 2\phi }$	Vertical astigmatism
${\displaystyle Z_{3}^{-3}}$	06	09	10	11	3	−3	${\displaystyle {\sqrt {8}}\rho ^{3}\sin 3\phi }$	Vertical trefoil
${\displaystyle Z_{3}^{-1}}$	07	07	07	08	3	−1	${\displaystyle {\sqrt {8}}(3\rho ^{3}-2\rho )\sin \phi }$	Vertical coma
${\displaystyle Z_{3}^{1}}$	08	08	06	07	3	+1	${\displaystyle {\sqrt {8}}(3\rho ^{3}-2\rho )\cos \phi }$	Horizontal coma
${\displaystyle Z_{3}^{3}}$	09	10	09	10	3	+3	${\displaystyle {\sqrt {8}}\rho ^{3}\cos 3\phi }$	Oblique trefoil
${\displaystyle Z_{4}^{-4}}$	10	15	17	18	4	−4	${\displaystyle {\sqrt {10}}\rho ^{4}\sin 4\phi }$	Oblique quadrafoil
${\displaystyle Z_{4}^{-2}}$	11	13	12	13	4	−2	${\displaystyle {\sqrt {10}}(4\rho ^{4}-3\rho ^{2})\sin 2\phi }$	Oblique secondary astigmatism
${\displaystyle Z_{4}^{0}}$	12	11	08	09	4	00	${\displaystyle {\sqrt {5}}(6\rho ^{4}-6\rho ^{2}+1)}$	Primary spherical
${\displaystyle Z_{4}^{2}}$	13	12	11	12	4	+2	${\displaystyle {\sqrt {10}}(4\rho ^{4}-3\rho ^{2})\cos 2\phi }$	Vertical secondary astigmatism
${\displaystyle Z_{4}^{4}}$	14	14	16	17	4	+4	${\displaystyle {\sqrt {10}}\rho ^{4}\cos 4\phi }$	Vertical quadrafoil