Zernike polynomials and their applications

When Zernike first introduced the orthogonal polynomials, radial polynomials and azimuthal functions were explicitly given [1, 2]. However, the normalization and ordering methods for these polynomials were not specified. Noll in 1976 sorted and normalized Zernike polynomials to facilitate statistical analysis of wavefront distortion caused by atmospheric turbulence [9]. The indexing scheme was later followed by many authors [34–36] and was used in commercial software, such as Zemax [25] as the standard indices. Note that the 'standard' indices in Zemax are not associated with any ANSI or ISO standards. In this section, we summarize the Noll indexing scheme, discuss normalized and non-normalized Zernike circle polynomials, and extend the definitions from the real domain to the complex domain.

2.1.1.1. Real Zernike circle polynomials.

In the Noll indices, the normalized or orthonormal Zernike circle polynomials are defined as the products of normalization factors, radial polynomials, and azimuthal (angular) functions, which are written as [9]:

$$\begin{align} {Z_j}(\rho ,\theta ) & = Z_n^m(\rho ,\theta ) \nonumber\\ & = \left\{ {\begin{array}{*{20}{l}} {\sqrt {2(n + 1)} R_n^m(\rho )\cos m\theta ,\ }{m \ne 0, \,\,j {\text{ is even}},} \\ {\sqrt {2(n + 1)} R_n^m(\rho )\sin m\theta ,\ }{m \ne 0,\,\, j {\text{ is odd,}}} \\ {\sqrt {(n + 1)} R_n^m(\rho ),\ }{m = 0,} \end{array} } \right.\end{align} \tag{ 3 }$$

where the index n is the degree of the radial polynomials, $R_n^m(\rho )$; the index m is the azimuthal frequency describing the repetition of the angular function; n and m are non-negative integers and satisfy n − m⩾ 0 and n − m= even; j is a mode-ordering number starting from 1 and its relationships with n and m are presented in equation (8). There are a total of (n + 1)(n + 2)/2 linearly independent polynomials for a degree ⩽ n. The radial polynomial $R_n^m(\rho )$ is defined as [1, 6]:

$$\begin{equation}R_n^m(\rho ) = \sum\limits_{s = 0}^{(n - m)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!\left( {\frac{{n + m}}{{\text{2}}} - s} \right)!\left( {\frac{{n - m}}{{\text{2}}} - s} \right)!}}} {\rho ^{n - 2s}}.\end{equation} \tag{ 4 }$$

The radial polynomials of the first few degrees are shown in figure 6. It is easy to verify the following relations:

$$\begin{equation}R_n^m(1) = 1,\end{equation} \tag{ 5 }$$

$$\begin{equation}\left| {{Z_j}(\rho ,\theta )} \right| \leqslant \sqrt {\frac{{2(n + 1)}}{{1 + {\delta _{m0}}}}} .\end{equation} \tag{ 6 }$$

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The normalized Zernike circle polynomials meet the following orthonormality condition:

$$\begin{equation}\frac{{\int_0^{2\pi } {\int_0^1 {{Z_j}} } (\rho ,\theta ){Z_{j^{\prime}}}(\rho ,\theta )\rho {\text{d}}\rho {\text{d}}\theta }}{{\int_0^{2\pi } {\int_0^1 {\rho {\text{d}}\rho {\text{d}}\theta } } }} = {\delta _{jj^{\prime}}},\end{equation} \tag{ 7 }$$

where ${\delta _{jj^{\prime}}}$ is the Kronecker delta function.

The orthonormal Zernike circle polynomials can be sorted by either the single index, j, or the double indices, n and m. The former is useful for describing Zernike expansion coefficients while the latter is useful for unambiguously describing the functions. To convert a given value j to n and m, one can use the following relationships [36]:

$$\begin{align} n & = \left\lfloor {\left( {\sqrt {2j - 1} + 0.5} \right) - 1} \right\rfloor , \nonumber\\[4pt] m & = \left\{ {\begin{array}{*{20}{l}} {2 \times \left\lfloor {\frac{{2j + 1 - n(n + 1)}}{4}} \right\rfloor ,}&{n{\text{ is even,}}} \\[6pt] {2 \times \left\lfloor {\frac{{2(j + 1) - n(n + 1)}}{4}} \right\rfloor - 1,}&{n{\text{ is odd,}}} \end{array}} \right. \end{align} \tag{ 8 }$$

where ⌊x⌋ denotes the floor function that gives as output the greatest integer less than or equal to x. For example, ⌊2.4⌋ = 2. To convert given values of n and m to j, the following relationship can be used:

$$\begin{align}j = \left\{ {\begin{array}{*{20}{l}} {\frac{{n(n + 1)}}{2} + 1,}&{m = 0,} \\[3pt] {\left[ {\frac{{n(n + 1)}}{2} + m,\frac{{n(n + 1)}}{2} + m + 1} \right],}&{m \ne 0.} \end{array}} \right.\end{align} \tag{ 9 }$$

Table 2 lists the first 37-term real orthonormal Zernike circle polynomials in the polar and Cartesian coordinate systems and the values for n, m, and j.

Table 2. First 37-term orthonormal Zernike circle polynomials under the Noll indices [25, 36].

j	n	m	Zj (ρ, θ)	Zj (x, y)	Aberration
1	0	0	1	1	Piston
2	1	1	$2\rho \cos \theta$	$2x$	x-tilt
3		1	$2\rho \sin \theta$	$2y$	y-tilt
4	2	0	$\sqrt 3 (2{\rho ^2} - 1)$	$\sqrt 3 [2({x^2} + {y^2}) - 1]$	Defocus
5		2	$\sqrt 6 {\rho ^2}\sin 2\theta$	$2\sqrt 6 xy$	45° Primary astigmatism
6		2	$\sqrt 6 {\rho ^2}\cos 2\theta$	$\sqrt 6 ({x^2} - {y^2})$	0° Primary astigmatism
7	3	1	$\sqrt 8 (3{\rho ^3} - 2\rho )\sin \theta$	$\sqrt 8 y[3({x^2} + {y^2}) - 2]$	Primary y-coma
8		1	$\sqrt 8 (3{\rho ^3} - 2\rho )\cos \theta$	$\sqrt 8 x[3({x^2} + {y^2}) - 2]$	Primary x-coma
9		3	$\sqrt 8 {\rho ^3}\sin 3\theta$	$\sqrt 8 y(3{x^2} - {y^2})$
10		3	$\sqrt 8 {\rho ^3}\cos 3\theta$	$\sqrt 8 x({x^2} - 3{y^2})$
11	4	0	$\sqrt 5 (6{\rho ^4} - 6{\rho ^2} + 1)$	$\sqrt 5 [6{({x^2} + {y^2})^2} - 6({x^2} + {y^2}) + 1]$	Primary spherical aberration
12		2	$\sqrt {10} (4{\rho ^4} - 3{\rho ^2})\cos 2\theta$	$\sqrt {10} ({x^2} - {y^2})[4({x^2} + {y^2}) - 3]$	0° Secondary astigmatism
13		2	$\sqrt {10} (4{\rho ^4} - 3{\rho ^2})\sin 2\theta$	$2\sqrt {10} xy[4({x^2} + {y^2}) - 3]$	45° Secondary astigmatism
14		4	$\sqrt {10} {\rho ^4}\cos 4\theta$	$\sqrt {10} [{({x^2} + {y^2})^2} - 8{x^2}{y^2}]$
15		4	$\sqrt {10} {\rho ^4}\sin 4\theta$	$4\sqrt {10} xy({x^2} - {y^2})$
16	5	1	$\sqrt {12} (10{\rho ^5} - 12{\rho ^3} + 3\rho )\cos \theta$	$\sqrt {12} x[10{({x^2} + {y^2})^2} - 12({x^2} + {y^2}) + 3]$	Secondary x-coma
17		1	$\sqrt {12} (10{\rho ^5} - 12{\rho ^3} + 3\rho )\sin \theta$	$\sqrt {12} y[10{({x^2} + {y^2})^2} - 12({x^2} + {y^2}) + 3]$	Secondary y-coma
18		3	$\sqrt {12} (5{\rho ^5} - 4{\rho ^3})\cos 3\theta$	$\sqrt {12} x({x^2} - 3{y^2})[5({x^2} + {y^2}) - 4]$
19		3	$\sqrt {12} (5{\rho ^5} - 4{\rho ^3})\sin 3\theta$	$\sqrt {12} y(3{x^2} - {y^2})[5({x^2} + {y^2}) - 4]$
20		5	$\sqrt {12} {\rho ^5}\cos 5\theta$	$\sqrt {12} x[16{x^4} - 20{x^2}({x^2} + {y^2}) + 5{({x^2} + {y^2})^2}]$
21		5	$\sqrt {12} {\rho ^5}\sin 5\theta$	$\sqrt {12} y[16{y^4} - 20{y^2}({x^2} + {y^2}) + 5{({x^2} + {y^2})^2}]$
22	6	0	$\sqrt 7 (20{\rho ^6} - 30{\rho ^4} + 12{\rho ^2} - 1)$	$\sqrt 7 [20{({x^2} + {y^2})^3} - 30{({x^2} + {y^2})^2} + 12({x^2} + {y^2}) - 1]$	Secondary spherical
23		2	$\sqrt {14} (15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\sin 2\theta$	$2\sqrt {14} xy[15{({x^2} + {y^2})^2} - 20({x^2} + {y^2}) + 6]$	45° Tertiary astigmatism
24		2	$\sqrt {14} (15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\cos 2\theta$	$\sqrt {14} ({x^2} - {y^2})[15{({x^2} + {y^2})^2} - 20({x^2} + {y^2}) + 6]$	0° Tertiary astigmatism
25		4	$\sqrt {14} (6{\rho ^6} - 5{\rho ^4})\sin 4\theta$	$4\sqrt {14} xy({x^2} - {y^2})[6({x^2} + {y^2}) - 5]$
26		4	$\sqrt {14} (6{\rho ^6} - 5{\rho ^4})\cos 4\theta$	$\sqrt {14} [{({x^2} + {y^2})^2} - 8{x^2}{y^2}][6({x^2} + {y^2}) - 5]$
27		6	$\sqrt {14} {\rho ^6}\sin 6\theta$	$\sqrt {14} xy[32{x^4} - 32{x^2}({x^2} + {y^2}) + 6{({x^2} + {y^2})^2}]$
28		6	$\sqrt {14} {\rho ^6}\cos 6\theta$	$\begin{gathered} \sqrt {14} [32{x^6} - 48{x^4}({x^2} + {y^2})\, + \\ 18{x^2}{({x^2} + {y^2})^2} - {({x^2} + {y^2})^3}] \\ \end{gathered}$
29	7	1	$4(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\sin \theta$	$4y[35{({x^2} + {y^2})^3} - 60{({x^2} + {y^2})^2} + 30({x^2} + {y^2}) - 4]$	Tertiary y-coma
30		1	$4(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\cos \theta$	$4x[35{({x^2} + {y^2})^3} - 60{({x^2} + {y^2})^2} + 30({x^2} + {y^2}) - 4]$	Tertiary x-coma
31		3	$4(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\sin 3\theta$	$4y(3{x^2} - {y^2})[21{({x^2} + {y^2})^2} - 30({x^2} + {y^2}) + 10]$
32		3	$4(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\cos 3\theta$	$4x({x^2} - 3{y^2})[21{({x^2} + {y^2})^2} - 30({x^2} + {y^2}) + 10]$
33		5	$4(7{\rho ^7} - 6{\rho ^5})\sin 5\theta$	$\begin{gathered} 4[4{x^2}y({x^2} - {y^2}) + y{({x^2} + {y^2})^2} - 8{x^2}{y^3}]\\ \times [7({x^2} + {y^2}) - 6]\qquad\quad\qquad\quad\quad\quad\;\; \\ \end{gathered}$
34		5	$4(7{\rho ^7} - 6{\rho ^5})\cos 5\theta$	$\begin{gathered} 4[x{({x^2} + {y^2})^2} - 8{x^3}{y^2} - 4x{y^2}({x^2} - {y^2})]\, \\ \times[7({x^2} + {y^2}) - 6]\qquad\quad\qquad\quad\quad\quad\;\; \\ \end{gathered}$
35		7	$4{\rho ^7}\sin 7\theta$	$\begin{gathered} 8{x^2}y[3{({x^2} + {y^2})^2} - 16{x^2}{y^2}]\,\;\quad\quad\;\; \\ + 4y({x^2} - {y^2})[{({x^2} + {y^2})^2} - 16{x^2}{y^2}] \\ \end{gathered}$
36		7	$4{\rho ^7}\cos 7\theta$	$\begin{gathered} 4x({x^2} - {y^2})[{({x^2} + {y^2})^2} - 16{x^2}{y^2}] \\ - 8x{y^2}[3{({x^2} + {y^2})^2} - 16{x^2}{y^2}]\;\;\quad \\ \end{gathered}$
37	8	0	$3(70{\rho ^8} - 140{\rho ^6} + 90{\rho ^4} - 20{\rho ^2} + 1)$	$\begin{gathered} 3[70{({x^2} + {y^2})^4} - 140{({x^2} + {y^2})^3}\,\quad \\ + 90{({x^2} + {y^2})^2} - 20({x^2} + {y^2}) + 1] \\ \end{gathered}$	Tertiary spherical

The non-normalized real Zernike circle polynomials can be obtained by dropping the normalization factors from the normalized Zernike circle polynomials as:

$$\begin{equation}{Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) = \left\{ {\begin{array}{*{20}{l}} {R_n^m(\rho )\cos m\theta ,\ }{m \ne 0,\,\, j {\text{ is even}},} \\[4pt] {R_n^m(\rho )\sin m\theta ,\ }{m \ne 0,\,\, j {\text{ is odd,}}} \\ {R_n^m(\rho ),\ }{m = 0.} \end{array} } \right.\end{equation} \tag{ 10 }$$

They satisfy the following relationship:

$$\begin{equation}\left| {{Z_j}(\rho ,\theta )} \right| \leqslant 1.\end{equation} \tag{ 11 }$$

The orthogonality of non-normalized Zernike circle polynomials can be written as:

$$\begin{equation}\frac{{\int_0^{2\pi } {\int_0^1 {{Z_j}(\rho ,\theta )} } {Z_{{j}{^{\prime}}}}(\rho ,\theta )\rho {\text{d}}\rho {\text{d}}\theta }}{{\int_0^{2{{\pi }}} {\int_0^1 {\rho {\text{d}}\rho {\text{d}}\theta } } }} = \frac{{1 + {\delta _{m0}}}}{{2(n + 1)}}{\delta _{jj^{\prime}}}.\end{equation} \tag{ 12 }$$

Note that the integral in the denominator is equal to π. Figures 7 and 8 show the three-dimensional (3D) visualization of the non-normalized Zernike circle polynomials up to the sixth degree and their corresponding interferometric fringe patterns as in optical testing [37].

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2.1.1.2. Complex Zernike circle polynomials.

The Zernike circle polynomials in the complex domain were not defined in Noll's original definition [9]. However, they can be obtained based on Bhatia and Wolf's work [3] by replacing the azimuthal functions in real Zernike circle polynomials with a complex exponential function. The orthonormal complex Zernike circle polynomials can be written as [4]:

$$\begin{equation}V\,_n^l(\rho ,\theta ) = \sqrt {n + 1} R_n^l(\rho )\exp ({\text{i}}l\theta ),\end{equation} \tag{ 13 }$$

where n is a non-negative integer, l is an integer, n − |l| ⩾ 0 and is even. The radial polynomial is defined as:

$$\begin{equation}R_n^l(\rho ) = \sum\limits_{s = 0}^{(n - |l|)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!\left( {\frac{{n + |l|}}{2} - s} \right)!\left( {\frac{{n - |l|}}{2} - s} \right)!}}} {\rho ^{n - 2s}}.\end{equation} \tag{ 14 }$$

The normalized complex Zernike circle polynomials meet the following orthonormality condition:

$$\begin{equation}\frac{{\int_0^{2\pi } {\int_0^1 {{{[V\,_n^l(\rho ,\theta )]}^*}} } V_{{n}{^{\prime}} }^{{l}{^{\prime}} }(\rho ,\theta )\rho {\text{d}}\rho {\text{d}}\theta }}{{\int_0^{2\pi } {\int_0^1 {\rho {\text{d}}\rho {\text{d}}\theta } } }} = {\delta _{ll^{\prime} }}{\delta _{{nn}{^{\prime}} }}.\end{equation} \tag{ 15 }$$

where * denotes complex conjugate.

The non-normalized version of the complex Zernike circle polynomials are defined as [32]:

$$\begin{equation}V\,_n^l(\rho ,\theta ) = R_n^l(\rho )\exp ({\text{i}}l\theta ).\end{equation} \tag{ 16 }$$

The orthogonality can be expressed as:

$$\begin{equation}{\textrm{ }}\frac{{\int_0^{2\pi } {\int_0^1 {{{[V{\mkern 1mu} _n^l(\rho ,\theta )]}^*}} } {V^{l\prime }}_{n\prime }(\rho ,\theta )\rho {\textrm{d}}\rho {\textrm{d}}\theta }}{{\int_0^{2\pi } {\int_0^1 {\rho {\textrm{d}}\rho {\textrm{d}}\theta } } }} = \frac{1}{{n + 1}}{\delta _{l{l^\prime }}}\delta {\textrm{ }}nn\prime .\end{equation} \tag{ 17 }$$

The complex and real definitions of Zernike circle polynomials are related via the Euler's formula [3, 32]. The complex version is useful to define Zernike moments in image analysis, which will be discussed in section 4.6.

The OSA/ANSI indices for Zernike circle polynomials were initially developed by an OSA Standards Taskforce in 1999 to reach consensus recommendations on definitions, conventions, and standards for reporting of optical aberrations of human eyes [26, 27]. It was later standardized in ANSI Z80.28 [13, 14] and ISO 24157 [15, 17] and adopted in some commercial software, such as COMSOL Ray Optics Module [38].

The Zernike circle polynomials in the OSA/ANSI indices employ a right-handed coordinate system, as shown in figure 9, and are defined in the real domain as [13, 14, 26, 27].

$$\begin{equation}{Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) = \left\{ {\begin{array}{*{20}{l}} {N_n^mR_n^{|m|}(\rho )\cos (m\theta ),}&{m \geqslant 0,} \\ {N_n^mR_n^{|m|}(\rho )\sin (\left| m \right|\theta ),}&{m < 0,} \end{array}} \right.\end{equation} \tag{ 18 }$$

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where n is a non-negative integer, m is an integer, n − |m| ⩾ 0 and is even, j is a mode-ordering number starting from 0. The radial polynomial, $R_n^m(\rho )$, is defined as:

$$\begin{equation}R_n^{|m|}(\rho ) = \sum\limits_{s = 0}^{(n - |m|)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!\left( {\frac{{n + \left| m \right|}}{2} - s} \right)!\left( {\frac{{n - \left| m \right|}}{2} - s} \right)!}}} {\rho ^{n - 2s}}.\end{equation} \tag{ 19 }$$

The normalization factor, $N_n^m$, can be written as:

$$\begin{equation}N_n^m = \sqrt {\frac{{2(n + 1)}}{{(1 + {\delta _{m0}})}}} .\end{equation} \tag{ 20 }$$

The Zernike circle polynomials under the OSA/ANSI indices can be sorted by either the single index, j, or the double indices, n and m. To achieve conversion among these indices, one can use the following relationships [26, 27]:

$$\begin{align} j & = \frac{{n(n + 2) + m}}{2}, \nonumber\\ n & = \left\lceil {\frac{{ - 3 + \sqrt {9 + 8j} }}{2}} \right\rceil , \nonumber\\ m & = 2j - n(n + 2), \end{align} \tag{ 21 }$$

where ⌈x⌉ denotes the ceiling function that gives as output the least integer greater than or equal to x. For example, when j = 4, n = ⌈1.7⌉ = 2, m = 0. Table 3 lists the first 37-term Zernike circle polynomials in the OSA/ANSI indices and the values for n, m, and j.

Table 3. First 37-term Zernike polynomials under the OSA/ANSI indices [14, 27, 39].

j	n	m	Zj	Aberration
0	0	0	1	Piston
1	1	−1	$2\rho \sin \theta$	y-tilt
2		1	$2\rho \cos \theta$	x-tilt
3	2	−2	$\sqrt 6 {\rho ^2}\sin 2\theta$	45° Primary astigmatism
4		0	$\sqrt 3 (2{\rho ^2} - 1)$	Defocus
5		2	$\sqrt 6 {\rho ^2}\cos 2\theta$	0° Primary astigmatism
6	3	−3	$\sqrt 8 {\rho ^3}\sin 3\theta$
7		−1	$\sqrt 8 (3{\rho ^3} - 2\rho )\sin \theta$	Primary y-coma
8		1	$\sqrt 8 (3{\rho ^3} - 2\rho )\cos \theta$	Primary x-coma
9		3	$\sqrt 8 {\rho ^3}\cos 3\theta$
10	4	−4	$\sqrt {10} {\rho ^4}\sin 4\theta$
11		−2	$\sqrt {10} (4{\rho ^4} - 3{\rho ^2})\sin 2\theta$	45° Secondary astigmatism
12		0	$\sqrt 5 (6{\rho ^4} - 6{\rho ^2} + 1)$	Primary spherical aberration
13		2	$\sqrt {10} (4{\rho ^4} - 3{\rho ^2})\cos 2\theta$	0° Secondary astigmatism
14		4	$\sqrt {10} {\rho ^4}\cos 4\theta$
15	5	−5	$\sqrt {12} {\rho ^5}\sin 5\theta$
16		−3	$\sqrt {12} (5{\rho ^5} - 4{\rho ^3})\sin 3\theta$
17		−1	$\sqrt {12} (10{\rho ^5} - 12{\rho ^3} + 3\rho )\sin \theta$	Secondary y-coma
18		1	$\sqrt {12} (10{\rho ^5} - 12{\rho ^3} + 3\rho )\cos \theta$	Secondary x-coma
19		3	$\sqrt {12} (5{\rho ^5} - 4{\rho ^3})\cos 3\theta$
20		5	$\sqrt {12} {\rho ^5}\cos 5\theta$
21	6	−6	$\sqrt {14} {\rho ^6}\sin 6\theta$
22		−4	$\sqrt {14} (6{\rho ^6} - 5{\rho ^4})\sin 4\theta$
23		−2	$\sqrt {14} (15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\sin 2\theta$	45° Tertiary astigmatism
24		0	$\sqrt 7 (20{\rho ^6} - 30{\rho ^4} + 12{\rho ^2} - 1)$	Secondary spherical
25		2	$\sqrt {14} (15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\cos 2\theta$	0° Tertiary astigmatism
26		4	$\sqrt {14} (6{\rho ^6} - 5{\rho ^4})\cos 4\theta$
27		6	$\sqrt {14} {\rho ^6}\cos 6\theta$
28	7	−7	$4{\rho ^7}\sin 7\theta$
29		−5	$4(7{\rho ^7} - 6{\rho ^5})\sin 5\theta$
30		−3	$4(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\sin 3\theta$
31		−1	$4(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\sin \theta$	Tertiary y-coma
32		1	$4(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\cos \theta$	Tertiary x-coma
33		3	$4(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\cos 3\theta$
34		5	$4(7{\rho ^7} - 6{\rho ^5})\cos 5\theta$
35		7	$4{\rho ^7}\cos 7\theta$
36	8	−8	$\sqrt {18} {\rho ^8}\sin 8\theta$

The Zernike circle polynomials under the fringe indexing scheme (also known as the USAF set) were first developed by John Loomis in an interferogram analysis program called FRINGE at the University of Arizona, Optical Sciences Center in the 1970s [10, 11, 40, 41]. They are a low-order Zernike set supplemented with radial polynomials of higher order and are preferred for lens design and optical metrology because they group terms according to optical wavefront aberration order [42, 43].

The Zernike circle polynomials under the fringe indices do not have normalization factors and can be written as:

$$\begin{equation}{Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) = \left\{ \begin{aligned} & R_n^{|m|}(\rho )\cos (m\theta ),\;m \geqslant 0, \\ & R_n^{|m|}(\rho )\sin (\left| m \right|\theta ),\;m < 0, \\ \end{aligned} \right.\end{equation} \tag{ 22 }$$

where n is a non-negative integer, m is an integer, n − |m| ⩾ 0 and is even, j is a mode-ordering number starting from 0 (In CODE V and Zemax, j starts from 1 instead of 0). The radial polynomial is expressed as:

$$\begin{equation}R_n^{|m|} = \sum\limits_{s = 0}^{(n - \left| m \right|)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!\left( {\frac{{n + \left| m \right|}}{{\text{2}}} - s} \right)!\left( {\frac{{n - \left| m \right|}}{{\text{2}}} - s} \right)!}}} {\rho ^{n - 2s}}.\end{equation} \tag{ 23 }$$

Note that the above formulas are modified from the Wyant and Creath formula [19] to facilitate comparisons with other indices. The final mathematical expression for each term, as listed in table 4, is the same as that in Wyant's notation.

Table 4. Fringe set of the Zernike circle polynomials [19, 25, 29].

j	N	n	m	Zj (ρ, θ)	Aberration
0	0	0	0	1	Piston
1	1	1	+1	$\rho \cos \theta$	Tilt X
2		1	−1	$\rho \sin \theta$	Tilt Y
3		2	0	$2{\rho ^2} - 1$	Defocus
4	2	2	+2	${\rho ^2}\cos 2\theta$	Astigmatism X
5		2	−2	${\rho ^2}\sin 2\theta$	Astigmatism Y
6		3	+1	$(3{\rho ^3} - 2\rho )\cos \theta$	Coma X
7		3	−1	$(3{\rho ^3} - 2\rho )\sin \theta$	Coma Y
8		4	0	$6{\rho ^4} - 6{\rho ^2} + 1$	Primary spherical
9	3	3	+3	${\rho ^3}\cos 3\theta$	Trefoil X
10		3	−3	${\rho ^3}\sin 3\theta$	Trefoil Y
11		4	+2	$(4{\rho ^4} - 3{\rho ^2})\cos 2\theta$	Secondary X astigmatism
12		4	−2	$(4{\rho ^4} - 3{\rho ^2})\sin 2\theta$	Secondary Y astigmatism
13		5	+1	$(10{\rho ^5} - 12{\rho ^3} + 3\rho )\cos \theta$	Secondary X coma
14		5	−1	$(10{\rho ^5} - 12{\rho ^3} + 3\rho )\sin \theta$	Secondary Y coma
15		6	0	$20{\rho ^6} - 30{\rho ^4} + 12{\rho ^2} - 1$	Secondary spherical
16	4	4	+4	${\rho ^4}\cos 4\theta$	Tetrafoil X
17		4	−4	${\rho ^4}\sin 4\theta$	Tetrafoil X
18		5	+3	$(5{\rho ^5} - 4{\rho ^3})\cos 3\theta$	Secondary X trefoil
19		5	−3	$(5{\rho ^5} - 4{\rho ^3})\sin 3\theta$	Secondary Y trefoil
20		6	+2	$(15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\cos 2\theta$	Tertiary X astigmatism
21		6	−2	$(15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\sin 2\theta$	Tertiary Y astigmatism
22		7	+1	$(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\cos \theta$	Tertiary X coma
23		7	−1	$(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\sin \theta$	Tertiary Y coma
24		8	0	$70{\rho ^8} - 140{\rho ^6} + 90{\rho ^4} - 20{\rho ^2} + 1$	Tertiary spherical
25	5	5	+5	${\rho ^5}\cos 5\theta$	Pentafoil X
26		5	−5	${\rho ^5}\sin 5\theta$	Pentafoil Y
27		6	+4	$(6{\rho ^6} - 5{\rho ^4})\cos 4\theta$	Secondary X Tetrafoil
28		6	−4	$(6{\rho ^6} - 5{\rho ^4})\sin 4\theta$	Secondary Y Tetrafoil
29		7	+3	$(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\cos 3\theta$	Tertiary X Trefoil
30		7	−3	$(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\sin 3\theta$	Tertiary Y Trefoil
31		8	+2	$(56{\rho ^8} - 105{\rho ^6} + 60{\rho ^4} - 10{\rho ^2})\cos 2\theta$	Quaternary X astigmatism
32		8	−2	$(56{\rho ^8} - 105{\rho ^6} + 60{\rho ^4} - 10{\rho ^2})\sin 2\theta$	Quaternary Y astigmatism
33		9	+1	$(126{\rho ^9} - 280{\rho ^7} + 210{\rho ^5} - 60{\rho ^3} + 5\rho )\cos \theta$	Quaternary X coma
34		9	−1	$(126{\rho ^9} - 280{\rho ^7} + 210{\rho ^5} - 60{\rho ^3} + 5\rho )\sin \theta$	Quaternary Y coma
35		10	0	$252{\rho ^{10}} - 630{\rho ^8} + 560{\rho ^6} - 210{\rho ^4} + 30{\rho ^2} - 1$	Quaternary Spherical
36	6	12	0	$924{\rho ^{12}} - 2772{\rho ^{10}} + 3150{\rho ^8} - 1680{\rho ^6} + 420{\rho ^4} - 42{\rho ^2} + 1$

Defining N = (n + |m|)/2, Zernike fringe polynomials can be sorted as follows. First arrange N in ascending order from 0 to 6; then sort n in ascending order for a given value of N; finally organize m in descending order for given values of N and n. Compared with other Zernike sets, the Zernike fringe set is unique in the sense that it only has 37 terms (N ⩽ 6). This small polynomial set is useful for interferogram analysis and automatic lens design and is widely adopted in commercial optical software, such as Zemax [25], CODE V [29], OSLO [44, 45], and MetroPro [46].

The ISO-14999 indices were first published by ISO in the ISO/TR 14999–2 technical report in 2005 [28] for the description of wavefront in interferometric measurement of optical elements and optical system and then updated in 2019 [16].

The Zernike circle polynomials under the ISO-14999 indexing scheme do not have normalization factors and can be written as [16, 28]:

$$\begin{equation}{Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) = \left\{ \begin{aligned} & R_n^{|m|}(\rho )\cos (m\theta ),\;\;m \geqslant 0, \\ & R_n^{|m|}(\rho )\sin (\left| m \right|\theta ),\;m < 0, \\ \end{aligned} \right.\end{equation} \tag{ 24 }$$

where n is a non-negative integer, m is an integer, n − |m| ⩾ 0 and is even, j is a mode-ordering number starting from 0 (j = 0, 1, 2, ..., ∞). The radial polynomial is expressed as:

$$\begin{equation}R_n^{|m|} = \sum\limits_{s = 0}^{(n - \left| m \right|)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!\left( {\frac{{n + \left| m \right|}}{{\text{2}}} - s} \right)!\left( {\frac{{n - \left| m \right|}}{{\text{2}}} - s} \right)!}}} {\rho ^{n - 2s}}.\end{equation} \tag{ 25 }$$

Defining N = n + |m|, the Zernike circle polynomials under the ISO-14999 indices are sorted as follows. First arrange N in ascending order from 0 to ∞; then sort n in ascending order for a given value of N; finally organize m in descending order for given values of N and n. One may find that the ISO-14999 indices share almost the same definition as the fringe indices except that the former contains infinite terms. Actually, the fringe set is a subset of the ISO-14999 set, which is called the Extended Fringe Zernike Polynomials in CODE V [29]. Table 5 lists the first 37 terms of the ISO-14999 set and the values for n, m, N, and j.

Table 5. First 37-term Zernike circle polynomials under the ISO-14999 indices [16, 28].

j	N	n	m	Zj	Aberration
0	0	0	0	1	Piston
1	2	1	1	$\rho \cos \theta$	x-Tilt
2		1	−1	$\rho \sin \theta$	y-Tilt
3		2	0	$2{\rho ^2} - 1$	Defocus
4	4	2	2	${\rho ^2}\cos 2\theta$	0° Primary astigmatism
5		2	−2	${\rho ^2}\sin 2\theta$	45° Primary astigmatism
6		3	1	$(3{\rho ^3} - 2\rho )\cos \theta$	Primary x-coma
7		3	−1	$(3{\rho ^3} - 2\rho )\sin \theta$	Primary y-coma
8		4	0	$6{\rho ^4} - 6{\rho ^2} + 1$	Primary spherical aberration
9	6	3	3	${\rho ^3}\cos 3\theta$
10		3	−3	${\rho ^3}\sin 3\theta$
11		4	2	$(4{\rho ^4} - 3{\rho ^2})\cos 2\theta$	0° Secondary astigmatism
12		4	−2	$(4{\rho ^4} - 3{\rho ^2})\sin 2\theta$	45° Secondary astigmatism
13		5	1	$(10{\rho ^5} - 12{\rho ^3} + 3\rho )\cos \theta$	Secondary x-coma
14		5	−1	$(10{\rho ^5} - 12{\rho ^3} + 3\rho )\sin \theta$	Secondary y-coma
15		6	0	$20{\rho ^6} - 30{\rho ^4} + 12{\rho ^2} - 1$	Secondary spherical
16	8	4	4	${\rho ^4}\cos 4\theta$
17		4	−4	${\rho ^4}\sin 4\theta$
18		5	3	$(5{\rho ^5} - 4{\rho ^3})\cos 3\theta$
19		5	−3	$(5{\rho ^5} - 4{\rho ^3})\sin 3\theta$
20		6	2	$(15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\cos 2\theta$	0° Tertiary astigmatism
21		6	−2	$(15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\sin 2\theta$	45° Tertiary astigmatism
22		7	1	$(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\cos \theta$	Tertiary x-coma
23		7	−1	$(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\sin \theta$	Tertiary y-coma
24		8	0	$70{\rho ^8} - 140{\rho ^6} + 90{\rho ^4} - 20{\rho ^2} + 1$	Tertiary spherical
25	10	5	5	${\rho ^5}\cos 5\theta$
26		5	−5	${\rho ^5}\sin 5\theta$
27		6	4	$(6{\rho ^6} - 5{\rho ^4})\cos 4\theta$
28		6	−4	$(6{\rho ^6} - 5{\rho ^4})\sin 4\theta$
29		7	3	$(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\cos 3\theta$
30		7	−3	$(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\sin 3\theta$
31		8	2	$(56{\rho ^8} - 105{\rho ^6} + 60{\rho ^4} - 10{\rho ^2})\cos 2\theta$	0° Quaternary astigmatism
32		8	−2	$(56{\rho ^8} - 105{\rho ^6} + 60{\rho ^4} - 10{\rho ^2})\sin 2\theta$	45° Quaternary astigmatism
33		9	1	$(126{\rho ^9} - 280{\rho ^7} + 210{\rho ^5} - 60{\rho ^3} + 5\rho )\cos \theta$	Quaternary x-coma
34		9	−1	$(126{\rho ^9} - 280{\rho ^7} + 210{\rho ^5} - 60{\rho ^3} + 5\rho )\sin \theta$	Quaternary y-coma
35		10	0	$252{\rho ^{10}} - 630{\rho ^8} + 560{\rho ^6} - 210{\rho ^4} + 30{\rho ^2} - 1$	Quaternary spherical
36	12	6	6	${\rho ^6}\cos 6\theta$

In the classic textbook Principle of Optics, Born and Wolf reviewed the definition of Zernike circle polynomials and used it for the expansion of aberration functions [32, 47]. Many people [12, 48] later follow the Born and Wolf definition and treat it as the 'standard' indexing scheme. However, as pointed out in Born and Wolf's book (appendix VII in the 7th expanded edition [32]), the indices actually originated from Bhatia and Wolf's work published in 1954 [3].

The Zernike circle polynomials under the Born and Wolf indices do not have normalization factors and can be written as [3, 32]:

$$\begin{equation}{Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) = \left\{ \begin{aligned} & R_n^{|m|}(\rho )\cos (m\theta ),\;\;m \geqslant 0, \\ & R_n^{|m|}(\rho )\sin (\left| m \right|\theta ),\;m < 0, \\ \end{aligned} \right.\end{equation} \tag{ 26 }$$

where n is a non-negative integer, m is an integer, n − |m| ⩾ 0 and is even, j is a mode-ordering number starting from 1 (j = 1, 2, ..., ∞). The radial polynomial is expressed as:

$$\begin{equation}R_n^{|m|} = \sum\limits_{s = 0}^{(n - \left| m \right|)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!\left( {\frac{{n + \left| m \right|}}{{\text{2}}} - s} \right)!\left( {\frac{{n - \left| m \right|}}{{\text{2}}} - s} \right)!}}} {\rho ^{n - 2s}}.\end{equation} \tag{ 27 }$$

The Zernike circle polynomials under the Born and Wolf indices are sorted as follows. First arrange n in ascending order from 0 to ∞ and then sort m in descending order for a given value of n. The Born and Wolf indices are used by several authors [12, 48] and software [29]. For example, although the software CODE V does not explicitly define the standard Zernike polynomials, the tabulated polynomials in its manual [29] have the same expressions as those in the Born and Wolf indices. Table 6 lists the first 37-term Zernike polynomials under the Born and Wolf indices and the values for n, m, and j.

Table 6. First 37-term Zernike circle polynomials under the Born and Wolf indices [29].

j	n	m	Zj (ρ, θ)	Aberration
1	0	0	1	Piston
2	1	1	$\rho \cos \theta$	x-tilt
3		−1	$\rho \sin \theta$	y-tilt
4	2	2	${\rho ^2}\cos 2\theta$	0° Primary astigmatism
5		0	$2{\rho ^2} - 1$	Defocus
6		−2	${\rho ^2}\sin 2\theta$	45° Primary astigmatism
7	3	3	${\rho ^3}\cos 3\theta$
8		1	$(3{\rho ^3} - 2\rho )\cos \theta$	Primary x-coma
9		−1	$(3{\rho ^3} - 2\rho )\sin \theta$	Primary y-coma
10		−3	${\rho ^3}\sin 3\theta$
11	4	4	${\rho ^4}\cos 4\theta$
12		2	$(4{\rho ^4} - 3{\rho ^2})\cos 2\theta$	0° Secondary astigmatism
13		0	$6{\rho ^4} - 6{\rho ^2} + 1$	Primary spherical aberration
14		−2	$(4{\rho ^4} - 3{\rho ^2})\sin 2\theta$	45° Secondary astigmatism
15		−4	${\rho ^4}\sin 4\theta$
16	5	5	${\rho ^5}\cos 5\theta$
17		3	$(5{\rho ^5} - 4{\rho ^3})\cos 3\theta$
18		1	$(10{\rho ^5} - 12{\rho ^3} + 3\rho )\cos \theta$	Secondary x-coma
19		−1	$(10{\rho ^5} - 12{\rho ^3} + 3\rho )\sin \theta$	Secondary y-coma
20		−3	$(5{\rho ^5} - 4{\rho ^3})\sin 3\theta$
21		−5	${\rho ^5}\sin 5\theta$
22	6	6	${\rho ^6}\cos 6\theta$
23		4	$(6{\rho ^6} - 5{\rho ^4})\cos 4\theta$
24		2	$(15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\cos 2\theta$	0° Tertiary astigmatism
25		0	$20{\rho ^6} - 30{\rho ^4} + 12{\rho ^2} - 1$	Secondary spherical
26		−2	$(15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\sin 2\theta$	45° Tertiary astigmatism
27		−4	$(6{\rho ^6} - 5{\rho ^4})\sin 4\theta$
28		−6	${\rho ^6}\sin 6\theta$
29	7	7	${\rho ^7}\cos 7\theta$
30		5	$(7{\rho ^7} - 6{\rho ^5})\cos 5\theta$
31		3	$(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\cos 3\theta$
32		1	$(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\cos \theta$	Tertiary x-coma
33		−1	$(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\sin \theta$	Tertiary y-coma
34		−3	$(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\sin 3\theta$
35		−5	$(7{\rho ^7} - 6{\rho ^5})\sin 5\theta$
36		−7	${\rho ^7}\sin 7\theta$
37	8	8	${\rho ^8}\cos 8\theta$

Different from the aforementioned five indexing schemes, the Malacara indices use a different coordinate convention, where the polar angle, θ, is measured clockwise from the +y-axis. The Zernike circle polynomials under the Malacara indices do not have normalization factors and can be written as:

$$\begin{equation}{Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) = \left\{ \begin{aligned} & R_n^{\left| m \right|}(\rho )\sin (m\theta ),\;m > 0, \\ & R_n^{\left| m \right|}(\rho )\cos (m\theta ),\;m \leqslant 0, \\ \end{aligned} \right.\end{equation} \tag{ 28 }$$

where n is a non-negative integer, m is an integer, n − |m| ⩾ 0 and is even, j is a mode-ordering number starting from 1 (j = 1, 2, ..., ∞). The radial polynomial is expressed as:

$$\begin{equation}R_n^{\left| m \right|} = \sum\limits_{s = 0}^{(n - \left| m \right|)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!\left( {\frac{{n + \left| m \right|}}{{\text{2}}} - s} \right)!\left( {\frac{{n - \left| m \right|}}{{\text{2}}} - s} \right)!}}} {\rho ^{n - 2s}}.\end{equation} \tag{ 29 }$$

The Zernike circle polynomials in the Malacara indices have the same ordering scheme as the Born and Wolf indices and are sorted as follows. First arrange n in ascending order from 0 to ∞ and then sort m in descending order for a given value of n. The Malacara indices are mainly used in the first and second editions of the well-known book Optical Shop Testing [30, 31], the third edition of which, however, defines the Zernike circle polynomials under the Noll indexing scheme [36]. Table 7 lists the first 37-term Zernike circle polynomials under the Malacara indices and the values for n, m, and j.

Table 7. First 37-term Zernike circle polynomials under the Malacara indices [31].

j	n	m	Zj (ρ, θ)	Aberration
1	0	0	1	Piston
2	1	1	$\rho \sin \theta$	x-tilt
3		−1	$\rho \cos \theta$	y-tilt
4	2	2	${\rho ^2}\sin 2\theta$	0° Primary astigmatism
5		0	$2{\rho ^2} - 1$	Defocus
6		−2	${\rho ^2}\cos 2\theta$	45° Primary astigmatism
7	3	3	${\rho ^3}\sin 3\theta$
8		1	$(3{\rho ^3} - 2\rho )\sin \theta$	Primary x-coma
9		−1	$(3{\rho ^3} - 2\rho )\cos \theta$	Primary y-coma
10		−3	${\rho ^3}\cos 3\theta$
11	4	4	${\rho ^4}\sin 4\theta$
12		2	$(4{\rho ^4} - 3{\rho ^2})\sin 2\theta$	0° Secondary astigmatism
13		0	$6{\rho ^4} - 6{\rho ^2} + 1$	Primary spherical aberration
14		−2	$(4{\rho ^4} - 3{\rho ^2})\cos 2\theta$	45° Secondary astigmatism
15		−4	${\rho ^4}\cos 4\theta$
16	5	5	${\rho ^5}\sin 5\theta$
17		3	$(5{\rho ^5} - 4{\rho ^3})\sin 3\theta$
18		1	$(10{\rho ^5} - 12{\rho ^3} + 3\rho )\sin \theta$	Secondary x-coma
19		−1	$(10{\rho ^5} - 12{\rho ^3} + 3\rho )\cos \theta$	Secondary y-coma
20		−3	$(5{\rho ^5} - 4{\rho ^3})\cos 3\theta$
21		−5	${\rho ^5}\cos 5\theta$
22	6	6	${\rho ^6}\sin 6\theta$
23		4	$(6{\rho ^6} - 5{\rho ^4})\sin 4\theta$
24		2	$(15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\sin 2\theta$	0° Tertiary astigmatism
25		0	$20{\rho ^6} - 30{\rho ^4} + 12{\rho ^2} - 1$	Secondary spherical
26		−2	$(15{\rho ^6} - 20{\rho ^4} + 6{\rho ^2})\cos 2\theta$	45° Tertiary astigmatism
27		−4	$(6{\rho ^6} - 5{\rho ^4})\cos 4\theta$
28		−6	${\rho ^6}\cos 6\theta$
29	7	7	${\rho ^7}\sin 7\theta$
30		5	$(7{\rho ^7} - 6{\rho ^5})\sin 5\theta$
31		3	$(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\sin 3\theta$
32		1	$(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\sin \theta$	Tertiary x-coma
33		−1	$(35{\rho ^7} - 60{\rho ^5} + 30{\rho ^3} - 4\rho )\cos \theta$	Tertiary y-coma
34		−3	$(21{\rho ^7} - 30{\rho ^5} + 10{\rho ^3})\cos 3\theta$
35		−5	$(7{\rho ^7} - 6{\rho ^5})\cos 5\theta$
36		−7	${\rho ^7}\cos 7\theta$
37	8	8	${\rho ^8}\sin 8\theta$

The different indexing schemes of Zernike circle polynomials are compared and summarized in table 8 from the perspectives of coordinate system, normalization, and ordering strategy. In particular, the sorting of the polynomials under different indices is shown in figure 10 for the first few degrees. An illustration of the sources and applications of the six indices is presented in figure 11. For convenience, the Noll indices will be used in the remaining part of the article unless otherwise stated.

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Table 8. Comparison of different indices for Zernike circle polynomials in the real domain.

	Noll	OSA/ANSI	Fringe/U of Arizona	ISO-14999	Born and Wolf	Malacara
Sources	Noll, 1976 [9] Zemax Standard ZPs [25] Optical Shop Testing (3rd ed), 2007 [36]	OSA VSIA Standards Taskforce, 2000 [26] ANSI Z80.28 [13, 14] ISO 24157 [15, 17] COMSOL Ray Optics Module [38]	Loomis, 1970s [10, 49] Wyant, 1992 [19] Zemax Fringe ZPs a [25] CODE V Fringe ZPs a [29] OSLO b [44, 45] MetroPro [46]	ISO/TR 14999–2, 2005 [16, 28] CODE V Extended Fringe ZPs a [29]	Bhatia and Wolf, 1954 [3] Born and Wolf, 1964 [50] CODE V Standard ZPs [29]	Optical Shop Testing (1st & 2nd eds), 1978 and 1992 [30, 31]
Domain	Real	Real	Real	Real	Real	Real
Coordinate system	0 ⩽ ρ ⩽ 1, θ from +x axis anticlockwise	0 ⩽ ρ ⩽ 1, θ from +x axis anticlockwise	0 ⩽ ρ ⩽ 1, θ from +x axis anticlockwise	0 ⩽ ρ ⩽ 1, θ from +x axis anticlockwise	0 ⩽ ρ ⩽ 1, θ from +x axis anticlockwise	0 ⩽ ρ ⩽ 1, θ from +y axis clockwise
Definition	$\begin{array}{l} {Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) \\ = \left\{ {\begin{array}{l} N_n^mR_n^m\cos m\theta ,\;m \ne 0, j {\text{ is even}} \\ N_n^mR_n^m\sin m\theta ,\;m \ne 0, j {\text{ is odd}} \\ N_n^mR_n^m,\;m = 0 \\ \end{array}} \right. \\ \end{array}$	$\begin{array}{l} {Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) \\ = \left\{ {\begin{array}{l} N_n^mR_n^{|m|}(\rho )\cos (m\theta ),\;\;m \geqslant 0 \\ N_n^mR_n^{|m|}(\rho )\sin (\left| m \right|\theta ),\;m < 0 \\ \end{array}} \right. \\ \end{array}$	$\begin{array}{l} {Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) \\ = \left\{ {\begin{array}{l} R_n^{|m|}(\rho )\cos (m\theta ),\;\;m \geqslant 0 \\ R_n^{|m|}(\rho )\sin (\left| m \right|\theta ),\;m < 0 \\ \end{array}} \right. \\ \end{array}$	$\begin{array}{l} {Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) \\ = \left\{ {\begin{array}{l} R_n^{|m|}(\rho )\cos (m\theta ),\;\;m \geqslant 0 \\ R_n^{|m|}(\rho )\sin (\left| m \right|\theta ),\;m < 0 \\ \end{array}} \right. \\ \end{array}$	$\begin{array}{l} {Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) \\ = \left\{ {\begin{array}{l} R_n^{|m|}(\rho )\cos (m\theta ),\;\;m \geqslant 0 \\ R_n^{|m|}(\rho )\sin (\left| m \right|\theta ),\;m < 0 \\ \end{array}} \right. \\ \end{array}$	$\begin{array}{l} {Z_j}(\rho ,\theta ) = Z_n^m(\rho ,\theta ) \\ = \left\{ {\begin{array}{l} R_n^{|m|}(\rho )\sin (m\theta ),\;\;m > 0 \\ R_n^{|m|}(\rho )\cos (m\theta ),\;m \leqslant 0 \\ \end{array}} \right. \\ \end{array}$
Radial polynomial	$R_n^m = \sum\limits_{s = 0}^{(n - m)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!\left( {\genfrac{}{}{.3pt}{3}{{n + m}}{{\text{2}}} - s} \right)!\left( {\genfrac{}{}{.3pt}{3}{{n - m}}{{\text{2}}} - s} \right)!}}} {\rho ^{n - 2s}}$	$R_n^{|m|} = \sum\limits_{s = 0}^{(n - \left| m \right|)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!\left( {\genfrac{}{}{.3pt}{3}{{n + \left| m \right|}}{{\text{2}}} - s} \right)!\left( {\genfrac{}{}{.3pt}{3}{{n - \left| m \right|}}{{\text{2}}} - s} \right)!}}} {\rho ^{n - 2s}}$
Normalization	Yes	Yes	No	No	No	No
Normalization factor	$N_n^m = \sqrt {\frac{{2(n + 1)}}{{(1 + {\delta _{m0}})}}}$	$N_n^m = \sqrt {\frac{{2(n + 1)}}{{(1 + {\delta _{m0}})}}}$	—	—	—	—
Term number	Infinite	Infinite	37	Infinite	Infinite	Infinite
Double indices	n: non-negative integer m: non-negative integer n − m ⩾ 0 and even	n: non-negative integer m: integer n − |m| ⩾ 0 and even	n: non-negative integer m: integer n − |m| ⩾ 0 and even N = (n + |m|)/2	n: non-negative integer m: integer n−|m| ⩾ 0 and even N = n + |m|	n: non-negative integer m: integer n−|m| ⩾ 0 and even	n: non-negative integer m: integer n−|m| ⩾ 0 and even
Single index	j = 1, 2, 3, ..., ∞	j = 0, 1, 2, ..., ∞	j = 0, 1, 2, ..., 36	j = 0, 1, 2, ..., ∞	j = 1, 2, 3, ..., ∞	j = 1, 2, 3, ..., ∞
Indexing scheme	n from 0 to ∞ m in ascending order	n from 0 to ∞ m in ascending order	N from 0 to 6 n in ascending order m in descending order	N from 0 to ∞ n in ascending order m in descending order	n from 0 to ∞ m in descending order	n from 0 to ∞ m in descending order
Major applications	General optics	Ophthalmic optics	Optical metrology	Optical metrology	Optical design	Rarely used

^a In CODE V and Zemax, the single index j for the fringe Zernike polynomials (ZPs) starts from 1. ^b In OSLO, the polar angle θ is measured from the +y axis [44].

The orthogonal relationships of real and complex Zernike circle polynomials have been presented and can be found in equations (7), (12), (15) and (17). Moreover, the radial and azimuthal functions of Zernike circle polynomials are also orthogonal and satisfy the following relationships [36]:

$$\begin{equation}\int_0^1 {R_n^m(\rho )} R_{{n}{^{\prime}}}^m(\rho )\rho {\text{d}}\rho = \frac{1}{{2(n + 1)}}{\delta _{{nn}{^{\prime}}}},\end{equation} \tag{ 30 }$$

$$\begin{equation}\begin{gathered} \int_0^{2{{\pi }}} {\left\{ \begin{aligned} & \cos m\theta \cos m^{\prime}\theta ,\;\;j\;{\text{and}}\;j^{\prime}\;{\text{are}}\;{\text{both even}}\;\; \\ & \cos m\theta \sin m^{\prime}\theta ,\;\;j\;{\text{is even and}}\;j^{\prime}\;{\text{is}}\;{\text{odd}} \\ & \sin m\theta \cos m^{\prime}\theta ,\;\;j\;{\text{is odd and}}\;j^{\prime}\;{\text{is}}\;{\text{even}} \\ & \sin m\theta \sin m^{\prime}\theta ,\;\;j\;{\text{and}}\;j^{\prime}\;{\text{are}}\;{\text{both odd}} \\ \end{aligned} \right.{\text{d}}\theta } \\ = \left\{ \begin{gathered} \pi (1 + {\delta _{m0}}){\delta _{mm^{\prime}}},\;j\;{\text{and}}\;j^{\prime}\;{\text{are}}\;{\text{both even;}} \\ \!\!\pi {\delta _{mm^{\prime}}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;j\;{\text{and}}\;j^{\prime}\;{\text{are}}\;{\text{both odd;}} \\ 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\quad\quad\;\;\;\;{\text{otherwise}}{\text{.}} \\ \end{gathered} \right. \\ \end{gathered} \end{equation} \tag{ 31 }$$

Note that the Noll indices are used here.

The symmetry of Zernike circle polynomials can be expressed as:

$$\begin{equation}Z_n^m(\rho ,\theta ) = {( - 1)^m}Z_n^m(\rho ,\theta + {{\pi }}).\end{equation} \tag{ 32 }$$

Define the Fourier transform pair as:

$$\begin{equation}{\mathcal{Z}}_j(u,v) = \iint {{Z_j}(x,y)\exp \left[ { - {\text{i}}2{{\pi }}\left( {ux + vy} \right)} \right]}{\text{d}}x{\text{d}}y,\end{equation} \tag{ 33 }$$

$$\begin{equation}{Z_j}(x,y) = \iint {{\mathcal{Z}_j}(u,v)\exp \left[ {{\text{i}}2{{\pi }}\left( {ux + vy} \right)} \right]}{\text{d}}u{\text{d}}v,\end{equation} \tag{ 34 }$$

where ${\mathcal{Z}_j}$ is the Fourier transform of Z_j and (u, v) are Cartesian coordinates in the frequency domain. Use (r, φ) to denote the polar coordinates in the frequency domain and apply the transformation relationships x = ρcosθ, y = ρsinθ, u = rcosφ, v = rsinφ, the Fourier transform of Zernike circle polynomials can be written as [9, 53]:

$$\begin{align} & {\mathcal{Z}_j}(r,\phi )\nonumber\\ & \quad = \int_0^{2{{\pi }}} {\int_0^1 {{Z_j}} } (\rho ,\theta )\exp \left[ { - {\text{i}}2{{\pi }}r\rho \cos (\theta - \phi )} \right]\rho {\text{d}}\rho {\text{d}}\theta \nonumber\\ & \quad = \left\{ {\begin{array}{*{20}{l}} {\sqrt {2(n + 1)} {{( - 1)}^{n/2 - m}}\frac{{{J_{n + 1}}(2{{\pi }}r)}}{r}\cos (m\phi ),}&{m \ne 0,\;j {\text{ is }} {\text{even}},} \\[4pt] {\sqrt {2(n + 1)} {{( - 1)}^{n/2 - m}}\frac{{{J_{n + 1}}(2{{\pi }}r)}}{r}\sin (m\phi ),}&{m \ne 0,\;j {\text{ is }} {\text{odd}},} \\[4pt] {\sqrt {(n + 1)} {{( - 1)}^{n/2}}\frac{{{J_{n + 1}}(2{{\pi }}r)}}{r},}&{m = 0,} \end{array}} \right. \end{align} \tag{ 35 }$$

where J_n (x) is the nth-order Bessel function of the first kind and is defined as [54]:

$$\begin{equation}{J_n}(x) = \sum\limits_{s = 0}^\infty {\frac{{{{( - 1)}^s}}}{{s!(n + s)!}}{{\left( {\frac{x}{2}} \right)}^{n + 2s}}} .\end{equation} \tag{ 36 }$$

The Fourier transform of Zernike circle polynomials is useful for the conversion between Zernike coefficients and Fourier series coefficients of a wavefront [53].

Substituting equation (35) into the inverse Fourier transform of Zernike circle polynomials (equation (34)), an integral representation of Zernike radial polynomials can be obtained as [2, 6]:

$$\begin{equation}R_n^m(\rho ) = 2{{\pi (}} - 1{{\text{)}}^{\frac{{n - m}}{2}}}\int_0^\infty {{J_{n + 1}}(2{{\pi }}r){J_m}(2{{\pi }}r\rho )} {\text{ d}}r.\end{equation} \tag{ 37 }$$

This integral representation is useful for deriving a recurrence relation of the derivatives of Zernike radial polynomials [9].

The integral representation for the radial polynomials provides a good starting point for calculating derivatives. The derivatives of radial polynomials can be written in a recursion relation as [9]:

$$\begin{equation}\frac{{{\text{d}}R_n^{|m|}(\rho )}}{{{\text{d}}\rho }} = n\left[ {R_{n - 1}^{|m| - 1}(\rho ) + R_{n - 1}^{|m| + 1}(\rho )} \right] + \frac{{{\text{d}}R_{n - 2}^{|m|}(\rho )}}{{{\text{d}}\rho }}.\end{equation} \tag{ 38 }$$

In polar coordinate system, the partial derivatives of Zernike circle polynomials under the Noll indices with respect to x and y can be written as [18, 55]:

$$\begin{equation}\begin{gathered} \frac{{\partial {Z_j}(\rho ,\theta )}}{{\partial x}} = N_n^m\left[ {\frac{{{\text{d}}R_n^{{\text{|}}m{\text{|}}}(\rho )}}{{{\text{d}}\rho }}{\Theta ^{{\text{|}}m{\text{|}}}}(\theta )\cos \theta - \frac{{R_n^{{\text{|}}m{\text{|}}}(\rho )}}{\rho }\frac{{{\text{d}}{\Theta ^{{\text{|}}m{\text{|}}}}(\theta )}}{{{\text{d}}\theta }}\sin \theta } \right], \\[4pt] \frac{{\partial {Z_j}(\rho ,\theta )}}{{\partial y}} = N_n^m\left[ {\frac{{{\text{d}}R_n^{{\text{|}}m{\text{|}}}(\rho )}}{{{\text{d}}\rho }}{\Theta ^{{\text{|}}m{\text{|}}}}(\theta )\sin \theta + \frac{{R_n^{{\text{|}}m{\text{|}}}(\rho )}}{\rho }\frac{{{\text{d}}{\Theta ^{{\text{|}}m{\text{|}}}}(\theta )}}{{{\text{d}}\theta }}\cos \theta } \right], \\\\[-4pt] \end{gathered} \end{equation} \tag{ 39 }$$

where the normalization factor and the azimuthal function are:

$$\begin{equation}N_n^m = \sqrt {\frac{{2(n + 1)}}{{(1 + {\delta _{m0}})}}} ,\end{equation} \tag{ 40 }$$

$$\begin{equation}{\Theta ^m}(\theta ) = \left\{ {\begin{array}{*{20}{l}} {\cos |m|\theta ,}&{m \ne 0,\;j {\text{ is }} {\text{even}},} \\ {\sin |m|\theta ,}&{m \ne 0,\;j {\text{ is }} {\text{odd,}}} \\ {1,}&{m = 0.} \end{array}} \right.\end{equation} \tag{ 41 }$$

In Cartesian coordinate system, the partial derivatives of Zernike circle polynomials under the OSA/ANSI indices with respect to x and y can be written as [14]:

$$\begin{align} \frac{{\partial {Z_j}(x,y)}}{{\partial x}} & = (1 + {\delta _{m0}})\left[ {\sum\limits_{{n}{^{\prime}} = |m| + 1}^{n - 1} {(n^{\prime} + 1){Z_{{n}{^{\prime}}}}^{\frac{m}{{|m|}}(|m| + 1)}} } \right. \nonumber\\ & \quad + \left. {(1 - {\delta _{m0}})(1 - {\delta _{m, - 1}})\sum\limits_{{n}{^{\prime}} = |m| - 1}^{n - 1} {(n^{\prime} + 1){Z_{{n}{^{\prime}}}}^{\frac{m}{{|m|}}(|m| - 1)}} } \!\right]\!, \nonumber\\ \frac{{\partial {Z_j}(x,y)}}{{\partial y}} & = (1 + {\delta _{m0}})\frac{m}{{|m|}}\left[ {\sum\limits_{{n}{^{\prime}} = |m| + 1}^{n - 1} {(n^{\prime} + 1){Z_{{n}{^{\prime}}}}^{ - \frac{m}{{|m|}}(|m| + 1)}} } \right. \nonumber\\ & \quad - \left. {(1 - {\delta _{m0}})(1 - {\delta _{m1}})\sum\limits_{{n}{^{\prime}} = |m| - 1}^{n - 1} {(n^{\prime} + 1){Z_{{n}{^{\prime}}}}^{ - \frac{m}{{|m|}}(|m| - 1)}} } \right], \end{align} \tag{ 42 }$$

where n' increases with a step of 2 in the summations and in the case that (|m| + 1) is larger than (n − 1), the first summation term does not exist. The Cartesian derivatives of Zernike circle polynomials can also be obtained using recurrence relations, as reported in [55–57]. The derivatives of Zernike circle polynomials are useful for certain problems, such as ray tracing in optical design [58] and wavefront reconstruction in wavefront sensing [22, 24].

Computation of high-order of Zernike circle polynomials is necessary for some applications, such as Zernike moments-based image analysis. Although the radial polynomials are explicitly formulated (equation (19)), direct numerical computation suffers from the problem of low computational efficiency and possible cancellation errors [59–61]. To deal with these problems, various recurrence relations have been proposed for evaluating the radial polynomials [60–64]. Here we briefly review four widely-used recurrence methods, including the modified Kintner method [59, 65], the Prata's method [66], the q-recursive method [59], and the Shakibaei and Paramesran method [62].

The modified Kintner method was first proposed by Kintner in 1976 [65] and improved by Chong et al in 2003 [59] by adding recurrence relations for special cases when n − m = 0 and 2. The improved recurrence relation can be expressed as:

$$\begin{align}R_n^m = \left\{ {\begin{array}{l} {{\rho ^n}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\quad\quad\text{when}\;n - m = 0,} \\[3pt] {n{\rho ^n} - (n - 1){\rho ^{n - 2}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\quad\text{when}\;\;n - m = 2,} \\[3pt] {\frac{1}{{{k_1}}}\left[ {({k_2}{\rho ^2} + {k_3})R_{n - 2}^m(\rho ) + {k_4}R_{n - 4}^m(\rho )} \right],\text{when}\;n - m = 4,6, 8, \ldots ,} \\ \end{array}} \right.\end{align} \tag{ 43 }$$

where n and m are non-negative integers and satisfy n − m ⩾ 0 and n − m = even; the coefficients are given by:

$$\begin{align} {k_1} & = \frac{{\text{1}}}{{\text{2}}}(n + m)(n - m)(n - 2), \nonumber\\ {k_2} & = 2n(n - 1)(n - 2), \nonumber\\ {k_3} & = - {m^2}(n - 1) - n(n - 1)(n - 2), \nonumber\\ {k_4} & = - \frac{{\text{1}}}{{\text{2}}}n(n + m - 2)(n - m - 2). \end{align} \tag{ 44 }$$

The modified Kintner method is a degree-varying (n-varying) approach that computes radial polynomials at higher order from those at lower order for a fixed value of m.

The Prata method was proposed by Prata and Rusch in 1989 [66] and the recurrence relation can be written as:

$$\begin{equation}R_n^m(\rho ) = \left\{ {\begin{array}{*{20}{l}} {{\rho ^n},}&{{\text{when }} n - m = 0,} \\ {{k_1}\rho R_{n - 1}^{\left| {m - 1} \right|}(\rho ) + {k_2}R_{n - 2}^m(\rho ),}& {{\text{when }} n - m = 2,4,6, \ldots ,} \end{array}} \right.\end{equation} \tag{ 45 }$$

where,

$$\begin{equation}{k_1} = \frac{{2n}}{{m + n}},\;\;{k_2} = \frac{{m - n}}{{m + n}}.\end{equation} \tag{ 46 }$$

The q-recursive method was proposed by Chong et al [59] and the three-term recurrence relation can be written as:

$$\begin{equation}R_n^m(\rho ) = \left\{ {\begin{array}{l} {{\rho ^n},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\quad \;\text{when}\;n - m = 0,} \\[3pt] {n{\rho ^n} - (n - 1){\rho ^{n - 2}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{when}\;n - m = 2,} \\[3pt] {{k_1}R_n^{m + 4}(\rho ) + \left( {{k_2} + \frac{{{k_3}}}{{{\rho ^2}}}} \right)R_n^{m + 2}(\rho ),\;\text{when}\;n - m = 4,6,8, \cdots ,} \\ \end{array}} \right.\end{equation} \tag{ 47 }$$

where the coefficients are given by:

$$\begin{align} {k_1} & = \frac{{(m + 4)(m + 3)}}{2} - (m + 4){k_2} \nonumber\\ & \quad + \frac{{{k_3}(n + m + 6)(n - m - 4)}}{8}, \nonumber\\ {k_2} & = \frac{{{k_3}(n + m + 4)(n - m - 2)}}{{4(m + 3)}} + (m + 2), \nonumber\\ {k_3} & = - \frac{{4(m + 2)(m + 1)}}{{(n + m + 2)(n - m)}}. \end{align} \tag{ 48 }$$

Different from the modified Kintner method and the Prata method, the q-recursive method is an m-varying method that computes radial polynomials at lower m from those at higher m for a fixed radial order n.

The Shakibaei and Paramesran method uses a particularly simple recursion, in which a radial polynomial is expressed as a linear combination of three earlier computed radial polynomials as [62]:

$$\begin{equation}R_n^m(\rho ) = \rho \left[ {R_{n - 1}^{\left| {m - 1} \right|}(\rho ) + R_{n - 1}^{m + 1}(\rho )} \right] - R_{n - 2}^m(\rho ).\end{equation} \tag{ 49 }$$

The recursion can be initialized with the conditions $R_0^0(\rho ) = 1$ and $R_n^m(\rho ) \equiv 0$ when n < m. According to [67], the speed and accuracy of the recursion outperforms the Prata method and the q-recursive method in an image processing setting.

The mathematical properties of Zernike circle polynomials are summarized in table 9.

Table 9. Properties of Zernike circle polynomials and their radial polynomials.

Properties	Radial polynomials	Zernike circle polynomials
Commutativity a	$\left\langle {R_n^m,R_{{n}{^{\prime}}}^{{m}{^{\prime}}}} \right\rangle = \int_0^1 {R_n^mR_{{n}{^{\prime}}}^{{m}{^{\prime}}}{\text{d}}\rho } = \left\langle {R_{{n}{^{\prime}}}^{{m}{^{\prime}}},R_n^m} \right\rangle$	$\left\langle {{Z_i},{Z_j}} \right\rangle = \int_{\text{0}}^{{\text{2}}\pi } {\int_{\text{0}}^{\text{1}} {{Z_i}{Z_j}\rho {\text{d}}\rho {\text{d}}\theta } } = \left\langle {{Z_j},{Z_i}} \right\rangle$
Homogeneity b	$\left\langle {cR_n^m,R_{{n}{^{\prime}}}^{{m}{^{\prime}}}} \right\rangle = \int_0^1 {cR_n^mR_{{n}{^{\prime}}}^{{m}{^{\prime}}}{\text{d}}\rho } = c\left\langle {R_n^m,R_{{n}{^{\prime}}}^{{m}{^{\prime}}}} \right\rangle$	$\left\langle {c{Z_i},{Z_j}} \right\rangle = \int_{\text{0}}^{{\text{2}}\pi } {\int_{\text{0}}^{\text{1}} {c{Z_i}{Z_j}\rho {\text{d}}\rho {\text{d}}\theta } } = c\left\langle {{Z_i},{Z_j}} \right\rangle$
Distributivity	$\langle R_n^m,R_{{n^\prime }}^{{m^\prime }} + R_{n''}^{m''}\rangle = \langle R_n^m,R_{{n^\prime }}^{{m^\prime }}\rangle + \langle R_n^m,R_{n''}^{m''}\rangle$	$\left\langle {{Z_i},{Z_j} + {Z_k}} \right\rangle = \left\langle {{Z_i},{Z_j}} \right\rangle + \left\langle {{Z_i},{Z_k}} \right\rangle$
Zero mean value	—	${\overline Z _j} = \int_\Sigma {{Z_j}(\rho ,\theta )\rho {\text{d}}\rho {\text{d}}\theta } = 0, \;j \ne 0$
Orthogonality	$\int_0^1 {R_n^m\left( \rho \right)} R_{{n}{^{\prime}}}^m\left( \rho \right)\rho {\text{d}}\rho = \frac{1}{{2\left( {n + 1} \right)}}{\delta _{{nn}{^{\prime}}}}$	$\begin{array}{l} {\text{Non-normalized real ZPs:}} \\[4pt] \;\;\frac{1}{{{\pi }}}\int_0^{2{{\pi }}} {\int_0^1 {{Z_j}} } {Z_{{j}{^{\prime}}}}(\rho )\rho {\text{d}}\rho {\text{d}}\theta = \frac{{1 + {\delta _{m0}}}}{{2(n + 1)}}{\delta _{jj^{\prime}}} \\[4pt] {\text{Normalized real ZPs:}} \\[4pt] \;\;\frac{1}{{{\pi }}}\int_0^{2{{\pi }}} {\int_0^1 {{Z_j}} } {Z_{{j}{^{\prime}}}}(\rho )\rho {\text{d}}\rho {\text{d}}\theta = {\delta _{jj^{\prime}}} \\[4pt] {\text{Non - normalized complex ZPs:}} \\[4pt] \;\;\frac{1}{{{\pi }}}\int_0^{2{{\pi }}} {\int_0^1 {{{[V_n^m(\rho )]}^ * }} } V_{{n}{^{\prime}}}^{{m}{^{\prime}}}(\rho )\rho {\text{d}}\rho {\text{d}}\theta = \frac{1}{{n + 1}}{\delta _{mm^{\prime}}}{\delta _{{nn}{^{\prime}}}} \\[4pt] {\text{Normalized complex ZPs:}} \\[4pt] \;\;\frac{1}{{{\pi }}}\int_0^{2{{\pi }}} {\int_0^1 {{{[V_n^m(\rho )]}^ * }} } V_{{n}{^{\prime}}}^{{m}{^{\prime}}}(\rho )\rho {\text{d}}\rho {\text{d}}\theta = {\delta _{mm^{\prime}}}{\delta _{{nn}{^{\prime}}}} \\[4pt] \end{array}$
Symmetry	$R_n^m(\rho ) = R_n^{ - m}(\rho )$	$Z_n^m(\rho ,\theta ) = {( - 1)^m}Z_n^m(\rho ,\theta + {{\pi }})$
Fourier transform	—	$\begin{aligned} & {{\mathcal{Z}}_j}(k,\phi ) \\ & = \left\{ {\begin{array}{*{20}{l}} {\sqrt {2(n + 1)} {{( - 1)}^{n/2 - m}}\frac{{{J_{n + 1}}(2{{\pi }}k)}}{k}\cos (m\phi ),}&{m \ne 0,\;j {\text{ is }} {\text{even}}} \\[6pt] {\sqrt {2(n + 1)} {{( - 1)}^{n/2 - m}}\frac{{{J_{n + 1}}(2{{\pi }}k)}}{k}\sin (m\phi ),}&{m \ne 0,\;j {\text{ is }} {\text{odd}}} \\[6pt] {\sqrt {(n + 1)} {{( - 1)}^{n/2}}\frac{{{J_{n + 1}}(2{{\pi }}k)}}{k},}&{m = 0} \end{array}} \right. \\ \end{aligned}$
Integral representation	$R_n^m\left( \rho \right) = 2{{\pi }}{\left( { - 1} \right)^{\frac{{n - m}}{2}}}\int_0^\infty {{J_{n + 1}}\left( {2{{\pi }}k} \right){J_m}\left( {2{{\pi }}k\rho } \right)} {\text{ d}}k$	—
Derivative	$\frac{{{\text{d}}R_n^{|m|}(\rho )}}{{{\text{d}}\rho }} = n\left[R_{n - 1}^{|m| - 1}(\rho ) + R_{n - 1}^{|m| + 1}(\rho )\right] + \frac{{{\text{d}}R_{n - 2}^{|m|}(\rho )}}{{{\text{d}}\rho }}$	$\begin{gathered} \frac{{\partial {Z_j}(\rho ,\theta )}}{{\partial x}} = \left[ {\frac{{\partial R_n^m(\rho )}}{{\partial \rho }}{\Theta ^m}(\theta )\cos \theta - \frac{{R_n^m(\rho )}}{\rho }\frac{{\partial {\Theta ^m}(\theta )}}{{\partial \theta }}\sin \theta } \right] \\ \frac{{\partial {Z_j}(\rho ,\theta )}}{{\partial y}} = \left[ {\frac{{\partial R_n^m(\rho )}}{{\partial \rho }}{\Theta ^m}(\theta )\sin \theta + \frac{{R_n^m(\rho )}}{\rho }\frac{{\partial {\Theta ^m}(\theta )}}{{\partial \theta }}\cos \theta } \right] \\ \end{gathered}$
Recurrence relation	$\begin{gathered} R_n^m = \frac{1}{{{k_1}}}\left[ {({k_2}{\rho ^2} + {k_3})R_{n - 2}^m(\rho ) + {k_4}R_{n - 4}^m(\rho )} \right]\; \\ R_n^m(\rho ) = {k_1}\rho R_{n - 1}^{\left| {m - 1} \right|}(\rho ) + {k_2}R_{n - 2}^m(\rho )\quad\qquad\;\; \\ R_n^m(\rho ) = {k_1}R_n^{m + 4}(\rho ) + \left( {{k_2} + \frac{{{k_3}}}{{{\rho ^2}}}} \right)R_n^{m + 2}(\rho )\; \\ \end{gathered}$	—

^a Angle brackets denote the inner product of two functions. ^b c is a constant.

A wavefront function W defined over a unit circle can be represented by the linear combination of finite terms of Zernike circle polynomials as [31, 34, 68, 69]:

$$\begin{equation}W(R\rho ,\theta ) = \sum\limits_{j = 0}^J {{a_j}{Z_j}(\rho ,\theta )} ,\end{equation} \tag{ 50 }$$

where R is the radius of the pupil, 0 ⩽ ρ⩽ 1, J is the maximum number of terms of the polynomials, a_j is the expansion coefficients, and Z_j is the jth-term Zernike circle polynomial. The equation can be equivalently expressed in Cartesian coordinates as:

$$\begin{equation}W(x,y) = \sum\limits_{j = 0}^J {{a_j}{Z_j}(x,y)} .\end{equation} \tag{ 51 }$$

Written in discrete and matrix forms, equation (51) becomes:

$$\begin{equation}{\mathbf{Aa = W}},\end{equation} \tag{ 52 }$$

where,

$$\begin{align} {\mathbf{A}} &= \left[ {\begin{array}{*{20}{c}} {{Z_1}({x_1},{y_1})}& {{Z_2}({x_1},{y_1})}& \cdots & {{Z_J}({x_1},{y_1})} \\ {{Z_1}({x_2},{y_2})}& {{Z_2}({x_2},{y_2})}& \cdots & {{Z_J}({x_2},{y_2})} \\ \vdots & \vdots & \ddots & \vdots \\ {{Z_1}({x_K},{y_K})}& {{Z_2}({x_K},{y_K})}& \cdots & {{Z_J}({x_K},{y_K})} \end{array}} \right], \nonumber\\ {\mathbf{a}} &= \left[ {\begin{array}{*{20}{c}} {{a_1}} \\ {{a_2}} \\ \vdots \\ {{a_J}} \end{array}} \right], {\mathbf{W}} = \left[ {\begin{array}{*{20}{c}} {W({x_1},{y_1})} \\ {W({x_2},{y_2})} \\ \vdots \\ {W({x_K},{y_K})} \end{array}} \right], \end{align} \tag{ 53 }$$

where K is the total number of data points within the unit circle. Generally, equation (52) is an overdetermined linear system, where there are more equations (K) than unknowns (J). It can be written into the normal equation [34, 68]:

$$\begin{equation}{\mathbf{A}^{\text{T}}}\mathbf{A}{\mathbf{a}} = {\mathbf{A}^{\text{T}}}\mathbf{W}{\text{,}}\end{equation} \tag{ 54 }$$

where the superscript T denotes matrix transpose. The solution can be obtained by matrix inversion as:

$$\begin{equation}{\mathbf{a}} = {({\mathbf{A}^{\text{T}}}\mathbf{A})^{ - 1}}{\mathbf{A}^{\text{T}}}\mathbf{W}.\end{equation} \tag{ 55 }$$

Figure 12 shows an example illustrating circular wavefront decomposition using orthonormal 37-term Zernike circle polynomials under the Noll indices. The amplitude of each coefficient indicates the strength of corresponding aberrations (table 2).

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The Zernike-based wavefront fitting has several useful properties [18]. First, the truncation of the expansion of a wavefront does not change the expansion coefficients. In other words, the expansion coefficients are independent from each other:

$$\begin{equation}{a_j} = \frac{{\text{1}}}{{{\pi }}}\int_{\text{0}}^{{\text{2}}\pi } {\int_{\text{0}}^{\text{1}} {W(\rho ,\theta ){Z_j}\rho {\text{d}}\rho {\text{d}}\theta } } .\end{equation} \tag{ 56 }$$

Second, all Zernike terms except the piston term have a mean value of zero and, therefore, the mean value of a wavefront equals the piston coefficient, i.e.:

$$\begin{equation}\overline W (\rho ,\theta ) = \frac{1}{\pi }\int_\Sigma {W(\rho ,\theta )\rho {\text{d}}\rho {\text{d}}\theta } = {a_0}.\end{equation} \tag{ 57 }$$

Third, wavefront variance equals the sum of the square of each expansion coefficient, excluding the piston coefficient, i.e.:

$$\begin{align}{\sigma ^2} & = \frac{{\text{1}}}{\pi }\int_{\text{0}}^{{\text{2}}\pi } {\int_{\text{0}}^{\text{1}} {{{\left[ {W(\rho ,\theta ) - \overline W (\rho ,\theta )} \right]}^{\text{2}}}\rho {\text{d}}\rho {\text{d}}\theta } }\nonumber\\ & = \overline {{W^{\,2}}} - {\left( {\overline W } \right)^2} = \sum\limits_{j = 1}^\infty {a_j^2} .\end{align} \tag{ 58 }$$

The properties of Zernike based wavefront fitting are summarized in table 10.

Table 10. Properties of Zernike based wavefront fitting [18, 35, 36].

Property	Formula
Coefficients independence	${a_j} = \frac{{\text{1}}}{{{\pi }}}\int_{\text{0}}^{{\text{2}}\pi } {\int_{\text{0}}^{\text{1}} {W(\rho ,\theta ){Z_j}\rho {\text{d}}\rho {\text{d}}\theta } }$
Wavefront mean value	$\overline W (\rho ,\theta ) = \frac{{\text{1}}}{{{\pi }}}\int_{\text{0}}^{{\text{2}}\pi } {\int_{\text{0}}^{\text{1}} {W(\rho ,\theta )\rho {\text{d}}\rho {\text{d}}\theta } } = {a_0}$
Wavefront mean square value	$\overline {{W^{\,2}}} (\rho ,\theta ) = \frac{{\text{1}}}{{{\pi }}}\int_{\text{0}}^{{\text{2}}\pi } {\int_{\text{0}}^{\text{1}} {{W^{\,2}}(\rho ,\theta )\rho {\text{d}}\rho {\text{d}}\theta } } = \sum\limits_{j = 0}^\infty {a_j^2}$
Wavefront variance	${\sigma ^2} = \overline {{W^{\,2}}} - {(\overline W )^2} = \sum\limits_{j = 1}^\infty {a_j^2}$

Zernike polynomials and their associated coefficients are commonly used to quantify the wavefront aberrations of the eye. When the aberrations of different eyes, pupil sizes, or corrections are compared or averaged, it is important that the Zernike coefficients have been calculated for the correct position, orientation, and size of the pupil. In this section, we discuss transformation relationships of Zernike expansion coefficients for translated, rotated, and resized pupils, which are shown in figure 13.

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2.3.2.1. Translation.

Translating a pupil (figure 13(b)) changes the expansion coefficients of a wavefront defined over it. Assuming that the displacements along the x and the y axis are Δx and Δy, respectively, the translated wavefront function can be expanded using the Taylor series [14, 70] as:

$$\begin{align} & W(x - \Delta x,y - \Delta y)\nonumber\\ & \quad = \sum\limits_{s = 0}^\infty {\frac{{{{( - 1)}^s}}}{{s!}}{{\left( {\Delta x\frac{\partial }{{\partial x}} + \Delta y\frac{\partial }{{\partial y}}} \right)}^s}W(x,y)} \nonumber\\ & \quad = \sum\limits_{n,m} {a_n^m\sum\limits_{s = 0}^\infty {\frac{{{{( - 1)}^s}}}{{s!}}{{\left( {\Delta x\frac{\partial }{{\partial x}} + \Delta y\frac{\partial }{{\partial y}}} \right)}^s}Z_n^m(x,y)} } .\end{align} \tag{ 59 }$$

New wavefront expansion coefficients $b_n^m$ can be obtained by computing the first-order derivatives of the Zernike circle polynomials in equation (59), which can be expressed as linear combinations of the untransformed Zernike circle polynomials (see section 2.2.5).

2.3.2.2. Rotation.

Rotating a pupil (figure 13(c)) similarly changes the expansion coefficients of a wavefront defined over it, which should be taken into consideration for applications such as vision correction surgery [71]. For a wavefront counterclockwise rotated with respect to its original coordinate system by an angle α, transformed Zernike expansion coefficients $b_n^m$ can be derived from original expansion coefficients $a_n^m$ as [18]:

$$\begin{align} b_n^{ - |m|} & = a_n^{ - |m|}\cos (\left| m \right|\alpha ) + a_n^{|m|}sin(\left| m \right|\alpha ), \nonumber\\ b_n^{|m|} & = - a_n^{ - |m|}\sin (\left| m \right|\alpha ) + a_n^{|m|}\cos (\left| m \right|\alpha ). \end{align} \tag{ 60 }$$

2.3.2.3. Resizing.

Comparison of Zernike expansion coefficients of wavefronts over different non-normalized pupils requires the same aperture size. Therefore, it is necessary to calculate expansion coefficients for an arbitrary pupil size based on the expansion coefficients of the full pupil. Many transformation relationships for pupil resizing have been developed [72–81] and two simpler methods were described by Dai [18, 82] and Janssen [79, 83].

Suppose there are two wavefronts, W₁ and W₂, defined over concentric pupils with radii of R₁ and R₂, respectively, as shown in figures 13(a) and (d). W₂ is part of W₁ and R₂ ⩽ R₁. The Zernike expansion of the wavefront W₂ can be written as:

$$\begin{equation}{W_2}({R_2}\rho ,\theta ) = \mathop \sum \limits_{j = 0}^J {b_j}{Z_j}(\rho ,\theta ),\end{equation} \tag{ 61 }$$

where 0 ⩽ ρ⩽ 1, b_j is the expansion coefficients in the OSA/ANSI indices for the wavefront W₂. Define a scale factor = R₂/R₁ and the expansion can also be written as:

$$\begin{equation}{W_2}({R_2}\rho ,\theta ) = {W_1}(\varepsilon {R_1}\rho ,\theta ) = \mathop \sum \limits_{j = 0}^J {a_j}{Z_j}(\varepsilon \rho ,\theta ),\end{equation} \tag{ 62 }$$

where a_j is the expansion coefficients for the wavefront W₁. Connecting equations (61) and (62), Dai gives [18, 82]:

$$\begin{equation}{b_j} = b_n^m = \sum\limits_{i = 0}^{\left\lfloor {{{({n_{\max }} - n)} / 2}} \right\rfloor } {G_n^i(\varepsilon )} a_{n + 2i}^m,\end{equation} \tag{ 63 }$$

where n_max is the maximum radial degree of the Zernike circle polynomials and $G_n^i(\varepsilon )$ is a resizing factor, defined as [18]:

$$\begin{align}G_n^i(\varepsilon ) & = \sqrt {(n + 2i + 1)(n + 1)} \nonumber\\[4pt] & \quad \times \left[ {\sum\limits_{j = 0}^i {\frac{{{{( - 1)}^{i + j}}(n + i + j)!}}{{j!(n + j + 1)!(i - j)!}}} {\varepsilon ^{2j}}} \right]{\varepsilon ^n},\end{align} \tag{ 64 }$$

where i ⩽ ⌊(n_max − n)/2⌋ and ⌊x⌋ denotes the floor function that gives as output the greatest integer less than or equal to x. The results suggest that the transformed expansion coefficient $b_n^m$ is a linear combination of $a_n^m$ and more untransformed coefficients $a_n^m$ are involved for the calculation of transformed coefficients $b_n^m$ for lower degrees. Table 11 lists the expressions for the resizing factor $G_n^i(\varepsilon )$ and transformed expansion coefficients $b_n^m$ for n_max = 6.

Table 11. Resizing factor $G_n^i(\varepsilon )$ and transformed expansion coefficients $b_n^m$ for n_max = 6 [18].

nmax	n	i	$G_n^i(\varepsilon )$	$b_n^m$
6	0	0	1	$G_0^0(\varepsilon )a_0^0 + G_0^1(\varepsilon )a_2^0 + G_0^2(\varepsilon )a_4^0 + G_0^3(\varepsilon )a_6^0$
		1	$- \sqrt 3 (1 - {\varepsilon ^2})$
		2	$\sqrt 5 (1 - 3{\varepsilon ^2} + 2{\varepsilon ^4})$
		3	$- \sqrt 7 (1 - 6{\varepsilon ^2} + 10{\varepsilon ^4} - 5{\varepsilon ^6})$
	1	0	$\varepsilon$	$G_1^0(\varepsilon )a_1^m + G_1^1(\varepsilon )a_3^m + G_1^2(\varepsilon )a_5^m$
		1	$- 2\sqrt 2 (1 - {\varepsilon ^2})\varepsilon$
		2	$\sqrt 3 (3 - 8{\varepsilon ^2} + 5{\varepsilon ^4})\varepsilon$
	2	0	${\varepsilon ^2}$	$G_2^0(\varepsilon )a_2^m + G_2^1(\varepsilon )a_4^m + G_2^2(\varepsilon )a_6^m$
		1	$- \sqrt {15} (1 - {\varepsilon ^2}){\varepsilon ^2}$
		2	$\sqrt {21} (2 - 5{\varepsilon ^2} + 3{\varepsilon ^4}){\varepsilon ^2}$
	3	0	${\varepsilon ^3}$	$G_3^0(\varepsilon )a_3^m + G_3^1(\varepsilon )a_5^m$
		1	$- 2\sqrt 6 (1 - {\varepsilon ^2}){\varepsilon ^3}$
	4	0	${\varepsilon ^4}$	$G_4^0(\varepsilon )a_4^m + G_4^1(\varepsilon )a_6^m$
		1	$- \sqrt {35} (1 - {\varepsilon ^2}){\varepsilon ^4}$
	5	0	${\varepsilon ^5}$	$G_5^0(\varepsilon )a_5^m$
	6	0	${\varepsilon ^6}$	$G_6^0(\varepsilon )a_6^m$

In addition to the Dai's formula (equation (63)), a concise expression with an elegant proof is given by Janssen and Dirksen as [79, 83]:

$$\begin{equation}b_n^m(\varepsilon ) = \sum\limits_{{n}{^{\prime}}} {a_{{n}{^{\prime}}}^m\left[ {R_{{n}{^{\prime}}}^n(\varepsilon ) - R_{{n}{^{\prime}}}^{n + 2}(\varepsilon )} \right]} , n = m,m + 2, \ldots ,\end{equation} \tag{ 65 }$$

where n' = n, n + 2, ..., and $R_n^{n + 2} \equiv 0$. The Janssen and Dirksen expression is mathematically equivalent to the Dai's formula but has the advantages of simplicity and only involving the radial polynomials, Zernike which can provide better numerical stability for high radial degrees.

A simple numerical simulation is presented in figure 14 to demonstrate the idea of wavefront resizing. Figure 14(a) is the original wavefront defined over a 3 mm-radius pupil and its first 37 expansion coefficients, a_j , under the OSA/ANSI indices are shown in figure 14(b). Figure 14(c) illustrates the Zernike expansion coefficients b_j for the 2 mm-radius portion of the original wavefront based on the conversion relationship in equation (63). Figures 14(d) and (e) show the reconstructed wavefront using the transformed coefficients b_j and the ground truth, respectively. The wavefront difference map is displayed in figure 14(d). The simulation suggests that equation (63) is effective for Zernike expansion coefficients computation over an arbitrary pupil size in wavefront resizing.

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The performance of an optical system can be characterized by the deformation of the wavefront emerging from the exit pupil relative to its reference sphere, that is, wavefront aberration, as shown in figure 15. The wavefront aberration for a rotationally symmetric system can be expanded by a set of power series in the four variables (ρ, θ) (polar coordinates of the exit pupil) and (r, φ = 0) (polar coordinates of an image point on the image plane) as [19, 20, 84, 85]:

$$\begin{align} W(\rho ,\theta ) & = \sum\limits_{l,n,m} {{W_{lnm}}{r^l}{\rho ^n}{{(\cos \theta )}^m}} \nonumber\\ & = {W_{111}}r\rho \cos \theta + {W_{020}}{\rho ^2} \nonumber\\ & \quad + {W_{040}}{\rho ^4} + {W_{131}}r{\rho ^3}\cos \theta + {W_{222}}{r^2}{\rho ^2}{\cos ^2}\theta \nonumber\\ & \quad + {W_{220}}{r^2}{\rho ^2} + {W_{311}}{r^3}\rho \cos \theta + {\text{higher - order terms}}, \end{align} \tag{ 66 }$$

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where l is a non-negative integer describing the dependence of the given term upon the distance of the image point from the axis; n and m are two non-negative integers determining the type of aberration. The first two terms in equation (66) represent the transverse (W₁₁₁) and the longitudinal (W₀₂₀) focal shifts, respectively. The remaining aberration terms constrained by the relation l + n = 4 are called primary or Seidel aberrations, which include five monochromatic aberrations, namely, spherical aberration (W₀₄₀), coma (W₁₃₁), astigmatism (W₂₂₂), field curvature (W₂₂₀), and distortion (W₃₁₁). These aberrations are sometimes called third-order aberrations when referring to ray aberration, which can be obtained as the derivative of wavefront aberration. For a fixed image point, r is a constant and can be absorbed into the coefficients. Assuming the relative aperture and the size of the field to be such that higher-order terms can be ignored, the expression of the wavefront aberration in equation (66) reduces to [19, 20]:

$$\begin{align} W(\rho ,\theta ) & = {W_{11}}\rho \cos \theta + {W_{20}}{\rho ^2} \nonumber\\ & \quad + {W_{22}}{\rho ^2}{\,}{\cos ^2}\theta + {W_{31}}{\rho ^3}\cos \theta + {W_{40}}{\rho ^4}{\text{}} .\end{align} \tag{ 67 }$$

Table 12 lists the first- and third-order aberrations.

Table 12. First- and third- order aberrations [84].

l	n	m	Coefficients	Expressions	Aberrations
1	1	1	W111	$r\rho \cos \theta$	Transverse focal shift
0	2	0	W020	${\rho ^2}$	Longitudinal focal shift
2	2	2	W222	${r^2}{\rho ^2}{\cos ^2}\theta$	Astigmatism
0	4	0	W040	${\rho ^4}$	Spherical aberration
1	3	1	W131	$r{\rho ^3}\cos \theta$	Coma
2	2	0	W220	${r^2}{\rho ^2}$	Field curvature
3	1	1	W311	${r^3}\rho \cos \theta$	Distortion

The wavefront aberration for a rotationally symmetric system can also be expanded by a set of Zernike series instead of power series. Assuming the first nine terms of Zernike circle polynomials are used for the expansion, the wavefront aberration can be written as [20]:

$$\begin{align} W(\rho ,\theta ) & = \sum\limits_{j = 0}^8 {{a_j}{Z_j}(\rho ,\theta )} \nonumber\\ & = {a_0} + {a_1}\rho \cos \theta + {a_2}\rho \sin \theta + {a_3}(2{\rho ^2} - 1) \nonumber\\ & \quad + {a_4}{\rho ^2}\cos 2\theta + {a_5}{\rho ^2}\sin 2\theta + {a_6}(3{\rho ^3} - 2\rho )\cos \theta \nonumber\\ & \quad + {a_7}(3{\rho ^3} - 2\rho )\sin \theta + {a_8}(6{\rho ^4} - 6{\rho ^2} + 1) .\end{align} \tag{ 68 }$$

It can be further rearranged as [20]:

$$\begin{align} W(\rho ,\theta ) & = {a_p} + {a_t}\rho \cos (\theta - {\phi _t}) + {a_d}{\rho ^2} + {a_a}{\rho ^2}{\cos ^2}(\theta - {\phi _a}) \nonumber\\ & \quad + {a_c}{\rho ^3}\cos (\theta - {\phi _c}) + {a_s}{\rho ^4}, \end{align} \tag{ 69 }$$

wherein the expressions for the coefficients and phase angles are tabulated in table 13. The expansion in equation (69) has a similar form to equation (67) indicating the coefficients of Seidel aberrations can be converted from Zernike expansion coefficients. However, one should keep in mind that without considering field dependence, the terms in equation (69) are not true Seidel aberrations. Wavefront measurement using an interferometer only provides data at a single field point. For this reason, field curvature looks like defocus and distortion like tilt. A set of wavefronts from different object points should be measured to determine the Seidel aberrations unambiguously from a Zernike expansion.

Table 13. Relationship between Zernike coefficients and Seidel aberrations [20].

Aberrations	Coefficients	Phase
Piston	${a_p} = {a_0} - {a_3} + {a_8}$	—
Tilt	${a_t} = \sqrt {{{({a_1} - 2{a_6})}^2} + {{({a_2} - 2{a_7})}^2}}$	${\phi _t} = \arctan \left[ {({a_2} - 2{a_7})/({a_1} - 2{a_6})} \right]$
Defocus a	${a_d} = \left( {2{a_3} - 6{a_8} \pm \sqrt {a_4^2 + a_5^2} } \right)$	—
Astigmatism b	${a_a} = \mp 2\sqrt {a_4^2 + a_5^2}$	${\phi _a} = 1/2\arctan ({a_5}/{a_4})$
Coma	${a_c} = 3\sqrt {a_6^2 + a_7^2}$	${\phi _c} = \arctan ({a_7}{\text{/}}{{\text{a}}_6})$
Spherical	${a_s} = 6{a_8}$	—

^a The sign in the defocus coefficient is chosen to minimize the magnitude of the coefficient [20]. ^b The sign in the astigmatism coefficient is chosen to be opposite to the sign in the defocus coefficient [20].

The Strehl ratio is defined as the ratio of the intensity I at the Gaussian image point in the presence of aberration, divided by the intensity I₀ when no aberration was present, as shown in figure 16. It is given by [19]:

$$\begin{equation}{\text{Strehl ratio}} = \frac{I}{{{I_0}}} = \frac{1}{{{\pi ^2}}}{\left| {\int_0^{2\pi } {\int_0^1 {\exp \left[ {{\text{i}}2\pi W(\rho ,\theta )} \right]\rho {\text{d}}\rho {\text{d}}\theta } } } \right|^2},\end{equation} \tag{ 70 }$$

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where W is the wavefront aberration with respect to the best reference sphere in the unit of wavelength. The Strehl ratio is a good measure of image quality when an optical system is well corrected. For modest amounts of aberrations, equation (70) can be approximated as [86, 87]:

$$\begin{equation}{\text{Strehl ratio}} \approx \exp \left[ { - {{(2\pi \sigma )}^2}} \right],\end{equation} \tag{ 71 }$$

where σ² is the variance of the wavefront across the pupil and is defined as [19]:

$$\begin{align}{\sigma ^2} & = \frac{{\text{1}}}{\pi }\int_{\text{0}}^{{\text{2}}\pi } {\int_{\text{0}}^{\text{1}} {{{\left[ {W(\rho ,\theta ) - \overline W (\rho ,\theta )} \right]}^{\text{2}}}\rho {\text{d}}\rho {\text{d}}\theta } } \nonumber\\ & = \overline {{W^{\,2}}} - {\left( {\overline W } \right)^2} = \sum\limits_{j = 1}^\infty {a_j^2} .\end{align} \tag{ 72 }$$

The Strehl ratio is inversely proportional to the variance of a wavefront, which can be characterized by Zernike expansion coefficients.

XY monomials are power series in x and y and in Cartesian coordinates are defined as [88]:

$$\begin{equation}T\,_n^m(x,y) = {x\,^m}{y^{n - m}},\end{equation} \tag{ 73 }$$

where n and m are non-negative integers and n ⩾ m. The XY monomials are also frequently used for representing wavefront aberrations, largely because they are a simple and complete set of basis functions. However, they are less popular than Zernike polynomials, especially after the 1980s, due to their non-orthogonality [88]. The conversions of wavefront expansion coefficients based on XY monomials and Zernike polynomials have been discussed by several authors and can be found in [88, 89].

The Jacobi polynomials are a class of classical orthogonal polynomials and can be defined by Rodrigues formula as [51]:

$$\begin{align}{P_n}^{(\alpha ,\beta )}(x) & = \frac{{{{( - 1)}^n}}}{{{2^n}n!{{(1 - x)}^\alpha }{{(1 + x)}^\beta }}}\frac{{{{\text{d}}^n}}}{{{\text{d}}{x^n}}}\nonumber\\ & \quad \times \left[ {{{(1 - x)}^{n + \alpha }}{{(1 + x)}^{n + \beta }}} \right],\end{align} \tag{ 74 }$$

where α, β > −1. Their explicit expressions are given as [51]:

$$\begin{align} {P_n}^{(\alpha ,\beta )}(x) & = \frac{{(n + \alpha )!(n + \beta )!}}{{{2^n}}} \nonumber\\ & \quad \times \sum\limits_{s = 0}^n {\frac{1}{{s!(n + \alpha - s)!(n - s)!(\beta + s)!}}}\nonumber\\&\quad\times {(x - 1)^{n - s}}{(x + 1)^m} .\end{align} \tag{ 75 }$$

They are orthogonal with respect to the weight (1 − x)^α (1 + x)^β on the interval [−1, 1]:

$$\begin{align} & \int_{ - 1}^1 {{{(1 - x)}^\alpha }{{(1 + x)}^\beta }p_n^{(\alpha ,\beta )}(x)P_{{n}{^{\prime}}}^{(\alpha ,\beta )}(x){\text{d}}x} \nonumber\\ & \quad = \frac{{{2^{\alpha + \beta + 1}}}}{{2n + \alpha + \beta + 1}}\frac{{(n + \alpha )!(n + \beta )!}}{{n!(n + \alpha + \beta )!}}{\delta _{{nn}{^{\prime}}}} .\end{align} \tag{ 76 }$$

The Zernike radial polynomials are a special case of the Jacobi polynomials multiplied by ρ^m with [90]:

$$\begin{equation}R_n^m(\rho ) = {( - 1)^{\frac{{n - m}}{2}}}{\rho ^m}P_{(n - m)/2}^{(m,0)}(1 - 2{\rho ^2}).\end{equation} \tag{ 77 }$$

The first few terms of the Jacobi polynomials are illustrated in figure 17. For more information about the Jacobi Polynomials, one can refer to [51, 91].

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The Legendre polynomials, sometimes called Legendre functions of the first kind, are solutions to the Legendre differential equation. They are a special class of the Jacobi polynomials with α = β = 0 and can be defined by Rodrigues formula as [51]:

$$\begin{equation}{P_n}(x) = \frac{{\text{1}}}{{{2^n}n!}}\frac{{{{\text{d}}^n}}}{{{\text{d}}{x^n}}}{({x^2} - 1)^n}.\end{equation} \tag{ 78 }$$

Their explicit expressions are given as [51]:

$$\begin{equation}{P_n}(x) = \frac{1}{{{2^n}}}\sum\limits_{s = 0}^{\left\lfloor {n/2} \right\rfloor } {{{( - 1)}^s}} \frac{{(2n - 2s)!}}{{s!(n - {\text{s}})!(n - 2s)!}}{x^{n - 2s}}.\end{equation} \tag{ 79 }$$

The Legendre polynomials are orthogonal over the interval [−1, 1]:

$$\begin{equation}\int_{ - 1}^1 {{P_n}(x){P_{{n}{^{\prime}}}}(x) = \frac{2}{{2n + 1}}{\delta _{{nn}{^{\prime}}}}} .\end{equation} \tag{ 80 }$$

They relate to the Zernike radial polynomials via [21]:

$$\begin{equation}R_{2n}^0(\rho ) = {P_n}(2{\rho ^2} - 1).\end{equation} \tag{ 81 }$$

The first few terms of the Legendre polynomials are illustrated in figure 18. For more information about the Legendre Polynomials, one can refer to [51, 92].

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The nth-order Bessel function of the first kind is defined as [92]:

$$\begin{equation}{J_n}(x) = \sum\limits_{s = 0}^\infty {\frac{{{{( - 1)}^s}}}{{s!(n + s)!}}{{\left( {\frac{x}{2}} \right)}^{n + 2s}}} .\end{equation} \tag{ 82 }$$

They relate to the Zernike radial polynomials via [32]:

$$\begin{equation}\int_0^1 {R_n^m(\rho ){J_m}(x\rho )\rho {\text{d}}\rho = {{( - 1)}^{\frac{{n - m}}{2}}}\frac{{{J_{n + 1}}(x)}}{x}} ,\end{equation} \tag{ 83 }$$

which is of great importance for the reduction of the diffraction integral in the Nijboer–Zernike theory [6, 21]. The first few terms of the Bessel functions are illustrated in figure 19.

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The Chebyshev polynomials of the second kind and of degree n are defined as [51]:

$$\begin{equation}{U_n}(x) = \frac{{\sin \left[ {(n + 1)\operatorname{arc} \cos(x)} \right]}}{{\sin \left[ {\operatorname{arc} \cos(x)} \right]}}.\end{equation} \tag{ 84 }$$

They relate to the Radon transforms of Zernike radial polynomials $\Re _n^m$ via [93]:

$$\begin{equation}\Re _n^m(s) = \frac{2}{{n + 1}}\sqrt {1 - {s^2}} {U_n}(s).\end{equation} \tag{ 85 }$$

The equation (85) can be used to compute the Zernike radial polynomials for large values of the degree n [94]. The first few terms of the Chebyshev polynomials of the second kind are illustrated in figure 20.

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The pseudo Zernike polynomials (see table 14), first derived by Bhatia and Wolf in 1954 [3], are a set of polynomials orthogonal over a unit circle and analogous to complex Zernike circle polynomials. They are obtained by eliminating the condition n − |l| = even from the definition of the complex Zernike circle polynomials in equation (16). Specifically, the pseudo Zernike polynomials are defined as:

$$\begin{equation}\mathcal{V}_n^l(\rho ,\theta ) = \mathcal{R}_n^l(\rho ){\text{exp(i}}l\theta {\text{)}},\end{equation} \tag{ 86 }$$

Table 14. Pseudo Zernike polynomials up to the 5th degree.

j	n	l	νj (ρ, θ)
1	0	0	1
2	1	−1	$\rho \exp ( - {\text{i}}\theta )$
3		0	$3\rho - 2$
4		1	$\rho \exp ({\text{i}}\theta )$
5	2	−2	${\rho ^2}\exp ( - {\text{i}}2\theta )$
6		−1	$(5{\rho ^2} - 4\rho )\exp ( - {\text{i}}\theta )$
7		0	$10{\rho ^2} - 12\rho + 3$
8		1	$(5{\rho ^2} - 4\rho )\exp ({\text{i}}\theta )$
9		2	${\rho ^2}\exp ({\text{i}}2\theta )$
10	3	−3	${\rho ^3}\exp ( - {\text{i}}3\theta )$
11		−2	$(7{\rho ^3} - 6{\rho ^2})\exp ( - {\text{i}}2\theta )$
12		−1	$(21{\rho ^3} - 30{\rho ^2} + 10\rho )\exp ( - {\text{i}}\theta )$
13		0	$35{\rho ^3} - 60{\rho ^2} + 30\rho - 4$
14		1	$(21{\rho ^3} - 30{\rho ^2} + 10\rho )\exp ({\text{i}}\theta )$
15		2	$(7{\rho ^3} - 6{\rho ^2})\exp ({\text{i}}2\theta )$
16		3	${\rho ^3}\exp ({\text{i}}3\theta )$
17	4	−4	${\rho ^4}\exp ( - {\text{i}}4\theta )$
18		−3	$(9{\rho ^4} - 8{\rho ^3})\exp ( - {\text{i}}3\theta )$
19		−2	$(36{\rho ^4} - 56{\rho ^3} + 21{\rho ^2})\exp ( - {\text{i}}2\theta )$
20		−1	$(84{\rho ^4} - 168{\rho ^3} + 105{\rho ^2} - 20\rho )\exp ( - {\text{i}}\theta )$
21		0	$126{\rho ^4} - 280{\rho ^3} + 210{\rho ^2} - 60\rho + 5$
22		1	$(84{\rho ^4} - 168{\rho ^3} + 105{\rho ^2} - 20\rho )\exp ({\text{i}}\theta )$
23		2	$(36{\rho ^4} - 56{\rho ^3} + 21{\rho ^2})\exp ({\text{i}}2\theta )$
24		3	$(9{\rho ^4} - 8{\rho ^3})\exp ({\text{i}}3\theta )$
25		4	${\rho ^4}\exp ({\text{i}}4\theta )$
26	5	−5	${\rho ^5}\exp ( - {\text{i}}5\theta )$
27		−4	$(11{\rho ^5} - 10{\rho ^4})\exp ( - {\text{i}}4\theta )$
28		−3	$(55{\rho ^5} - 90{\rho ^4} + 36{\rho ^3})\exp ( - {\text{i}}3\theta )$
29		−2	$(165{\rho ^5} - 360{\rho ^4} + 252{\rho ^3} - 56{\rho ^2})\exp ( - {\text{i}}2\theta )$
30		−1	$(330{\rho ^5} - 840{\rho ^4} + 756{\rho ^3} - 280{\rho ^2} + 35\rho )\exp ( - {\text{i}}\theta )$
31		0	$462{\rho ^5} - 1260{\rho ^4} + 1260{\rho ^3} - 560{\rho ^2} + 105\rho - 6$
32		1	$(330{\rho ^5} - 840{\rho ^4} + 756{\rho ^3} - 280{\rho ^2} + 35\rho )\exp ({\text{i}}\theta )$
33		2	$(165{\rho ^5} - 360{\rho ^4} + 252{\rho ^3} - 56{\rho ^2})\exp ({\text{i}}2\theta )$
34		3	$(55{\rho ^5} - 90{\rho ^4} + 36{\rho ^3})\exp ({\text{i}}3\theta )$
35		4	$(11{\rho ^5} - 10{\rho ^4})\exp ({\text{i}}4\theta )$
36		5	${\rho ^5}\exp ({\text{i}}5\theta )$

where n is a nonnegative integer, l is an integer, and n − |l| ⩾ 0; the radial polynomials of pseudo Zernike polynomials can be written as:

$$\begin{equation}\mathcal{R}_n^l(\rho ) = \sum\limits_{s = 0}^{n - \left| l \right|} {\frac{{{{( - 1)}^s}(2n + 1 - s)!}}{{s!(n + \left| l \right| + 1 - s)!(n - \left| l \right| - s)!}}} {\rho ^{n - s}}.\end{equation} \tag{ 87 }$$

The relation between the pseudo Zernike radial polynomials (equation (87)) and the Zernike radial polynomials (equation (19)) is given by [3]:

$$\begin{equation}\mathcal{R}_n^l({\rho ^2}) = \frac{1}{\rho }R_{2n + 1}^{2l + 1}(\rho ).\end{equation} \tag{ 88 }$$

The first few terms of the pseudo Zernike radial polynomials are illustrated in figure 21. Pseudo Zernike polynomials can be used for wavefront sensing [83], and to define pseudo Zernike moments, which can generate moment invariants as shape descriptors for pattern recognition (section 4.6.2).

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3.2.1.1. Real Zernike annular polynomials

Real Zernike annular polynomials have normalized and non-normalized forms. The normalized form defined under the Noll indexing scheme can be written as [4, 35, 112]:

$$\begin{align} {Z_j}(\rho ,\theta ;\varepsilon ) & = Z_n^m(\rho ,\theta ;\varepsilon ) \nonumber\\ & = \left\{ {\begin{array}{*{20}{l}} {{{[2(n + 1)]}^{1/2}}R_n^m(\rho ;\varepsilon )\cos m\theta ,}\;{m \ne 0, \,\,j {\text{ is even}},} \\ {{{[2(n + 1)]}^{1/2}}R_n^m(\rho ;\varepsilon )\sin m\theta ,}\;\;{m \ne 0,\,\, j {\text{ is odd,}}} \\ {{{[(n + 1)]}^{1/2}}R_n^0(\rho ;\varepsilon ),}\;\;\quad\quad \quad\;\;{m = 0,} \end{array} } \right. \end{align} \tag{ 93 }$$

where the index n is the degree of the radial polynomials, $R_n^m(\rho ;\varepsilon )$; the index m is the azimuthal frequency describing the repetition of the angular function; n and m are non-negative integers and satisfy n − m⩾ 0 and n − m= even; j is a mode-ordering number starting from 1, and is the obscuration ratio. There are a total of (n + 1)(n + 2)/2 linearly independent polynomials for a specific degree of n. The radial polynomials $R_n^m(\rho ;\varepsilon )$ can be obtained by Gram–Schmidt orthogonalization and are given by:

$$\begin{align} & R_n^m(\rho ;\varepsilon ) \nonumber\\ & \quad = \omega _n^m\Bigg[ {R_n^m(\rho ) } - {\sum\limits_{i = 1}^{(n - m)/2} {(n - 2i + 1)\overline {R_n^m(\rho )R_{n - 2i}^m(\rho ;\varepsilon )} R_{n - 2i}^m(\rho ;\varepsilon )} } \Bigg], \end{align} \tag{ 94 }$$

where:

$$\begin{equation}\overline {R_n^m(\rho )R_{n - 2i}^m(\rho ;\varepsilon )} = \frac{2}{{1 - {\varepsilon ^2}}}\int_\varepsilon ^1 {R_n^m(\rho )R_{n - 2i}^m(\rho ;\varepsilon )\rho {\text{d}}\rho } ,\end{equation} \tag{ 95 }$$

and the weighting factor $\omega _n^m$ can be determined according to the orthogonality condition of the radial polynomials as:

$$\begin{align}\omega _n^m = {\left( {\frac{{\dfrac{{1 - {\varepsilon ^2}}}{{2(n + 1)}}}}{{\int_\varepsilon ^1 {{{\left[ {R_n^m(\rho ) - \sum\limits_{i = 1}^{(n - m)/2} {(n - 2i + 1)\overline {R_n^m(\rho )R_{n - 2i}^m(\rho ;\varepsilon )} R_{n - 2i}^m(\rho ;\varepsilon )} } \right]}^2}\rho {\text{d}}\rho } }}} \right)^{1/2}}.\end{align} \tag{ 96 }$$

Exemplary profiles of the radial polynomials are shown in figure 24. It is easy to verify that when = 0, Zernike annular polynomials reduce to circle polynomials. The normalized Zernike annular polynomials meet the following orthonormality condition:

$$\begin{equation}\frac{{\int_0^{2\pi } {\int_\varepsilon ^1 {{Z_j}(\rho ,\theta ;\varepsilon )} } {Z_{{j}{^{\prime}}}}(\rho ,\theta ;\varepsilon )\rho {\text{d}}\rho {\text{d}}\theta }}{{\int_0^{2\pi } {\int_\varepsilon ^1 {\rho {\text{d}}\rho {\text{d}}\theta } } }} = {\delta _{jj^{\prime}}},\end{equation} \tag{ 97 }$$

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where ${\delta _{jj^{\prime}}}$ is the Kronecker delta function.

Similar to Zernike circle polynomials, orthonormal Zernike annular polynomials can be sorted by either the single index, j, or the double indices, n and m. The former is useful for describing Zernike expansion coefficients while the latter is useful for unambiguously describing the functions. To convert between the indices n, m, and j, one can use the relationships described in equation (8) and (9). Table 16 lists the first 28-term orthonormal real Zernike annular polynomials in the polar coordinate system and the values for n, m, and j. For more terms up to the 45th, one can refer to the tables 5–7 in [104].

Table 16. Zernike annular polynomials under the Noll indices up to the sixth degree (n = 6).

j	n	m	$R_n^m(\rho ;\varepsilon )$	${Z_j}(\rho ,\theta ;\varepsilon )$	Aberration
1	0	0	1	$R_0^0(\rho ;\varepsilon )$	Piston
2	1	1	$\frac{\rho }{{{{(1 + {\varepsilon ^2})}^{1/2}}}}$	$2R_1^1(\rho ;\varepsilon )\cos \theta$	X Tilt
3		1		$2R_1^1(\rho ;\varepsilon )\sin \theta$	Y Tilt
4	2	0	$\frac{{2{\rho ^2} - 1 - {\varepsilon ^2}}}{{1 - {\varepsilon ^2}}}$	$\sqrt 3 R_2^0(\rho ;\varepsilon )$	Defocus
5		2	$\frac{{{\rho ^2}}}{{{{(1 + {\varepsilon ^2} + {\varepsilon ^4})}^{1/2}}}}$	$\sqrt 6 R_2^2(\rho ;\varepsilon )\sin 2\theta$	Primary Y astigmatism
6		2		$\sqrt 6 R_2^2(\rho ;\varepsilon )\cos 2\theta$	Primary X astigmatism
7	3	1	$\frac{{3(1 + {\varepsilon ^2}){\rho ^3} - 2(1 + {\varepsilon ^2} + {\varepsilon ^4})\rho }}{{(1 - {\varepsilon ^2}){{[(1 + {\varepsilon ^2})(1 + 4{\varepsilon ^2} + {\varepsilon ^4})]}^{1/2}}}}$	$\sqrt 8 R_3^1(\rho ;\varepsilon )\sin \theta$	Primary Y coma
8		1		$\sqrt 8 R_3^1(\rho ;\varepsilon )\cos \theta$	Primary X coma
9		3	$\frac{{{\rho ^3}}}{{{{(1 + {\varepsilon ^2} + {\varepsilon ^4} + {\varepsilon ^6})}^{1/2}}}}$	$\sqrt 8 R_3^3(\rho ;\varepsilon )\sin 3\theta$
10		3		$\sqrt 8 R_3^3(\rho ;\varepsilon )\cos 3\theta$
11	4	0	$\frac{{[6{\rho ^4} - 6(1 + {\varepsilon ^2}){\rho ^2} + 1 + 4{\varepsilon ^2} + {\varepsilon ^4}]}}{{{{(1 - {\varepsilon ^2})}^2}}}$	$\sqrt 5 R_4^0(\rho ;\varepsilon )$	Primary spherical
12		2	$\frac{{4{\rho ^4} - 3[(1 - {\varepsilon ^8})/(1 - {\varepsilon ^6})]{\rho ^2}}}{{{{\left\{ {{{(1 - {\varepsilon ^2})}^{ - 1}}\left[ {16(1 - {\varepsilon ^{10}}) - 15{{(1 - {\varepsilon ^8})}^2}/(1 - {\varepsilon ^6})} \right]} \right\}}^{1/2}}}}$	$\sqrt {10} R_4^2(\rho ;\varepsilon )\cos 2\theta$	Secondary X astigmatism
13		2		$\sqrt {10} R_4^2(\rho ;\varepsilon )\sin 2\theta$	Secondary Y astigmatism
14		4	$\frac{{{\rho ^4}}}{{{{(1 + {\varepsilon ^2} + {\varepsilon ^4} + {\varepsilon ^6} + {\varepsilon ^8})}^{1/2}}}}$	$\sqrt {10} R_4^4(\rho ;\varepsilon )\cos 4\theta$
15		4		$\sqrt {10} R_4^4(\rho ;\varepsilon )\sin 4\theta$
16	5	1	$\frac{{\left[ 10(1 + 4{\varepsilon ^2} + {\varepsilon ^4}){\rho ^5} - 12(1 + 4{\varepsilon ^2} + 4{\varepsilon ^4} + {\varepsilon ^6}){\rho ^3} + 3(1 + 4{\varepsilon ^2} + 10{\varepsilon ^4} + 4{\varepsilon ^6} + {\varepsilon ^8})\rho \right]}}{{{{(1 - {\varepsilon ^2})}^2}{{[(1 + 4{\varepsilon ^2} + {\varepsilon ^4})(1 + 9{\varepsilon ^2} + 9{\varepsilon ^4} + {\varepsilon ^6})]}^{1/2}}}}$	$\sqrt {12} R_5^1(\rho ;\varepsilon )\cos \theta$	Secondary X coma
17		1		$\sqrt {12} R_5^1(\rho ;\varepsilon )\sin \theta$	Secondary Y coma
18		3	$\frac{{5{\rho ^5} - 4[(1 - {\varepsilon ^{10}})/(1 - {\varepsilon ^8})]{\rho ^3}}}{{{{\left\{ {{{(1 - {\varepsilon ^2})}^{ - 1}}[25(1 - {\varepsilon ^{12}}) - 24{{(1 - {\varepsilon ^{10}})}^2}/(1 - {\varepsilon ^8})]} \right\}}^{1/2}}}}$	$\sqrt {12} R_5^3(\rho ;\varepsilon )\cos 3\theta$
19		3		$\sqrt {12} R_5^3(\rho ;\varepsilon )\sin 3\theta$
20		5	$\frac{{{\rho ^5}}}{{{{(1 + {\varepsilon ^2} + {\varepsilon ^4} + {\varepsilon ^6} + {\varepsilon ^8} + {\varepsilon ^{10}})}^{1/2}}}}$	$\sqrt {12} R_5^5(\rho ;\varepsilon )\cos 5\theta$
21		5		$\sqrt {12} R_5^5(\rho ;\varepsilon )\sin 5\theta$
22	6	0	$\frac{{[20{\rho ^6} - 30(1 + {\varepsilon ^2}){\rho ^4} + 12(1 + 3{\varepsilon ^2} + {\varepsilon ^4}){\rho ^2} - (1 + 9{\varepsilon ^2} + 9{\varepsilon ^4} + {\varepsilon ^6})]}}{{{{(1 - {\varepsilon ^2})}^3}}}$	$\sqrt 7 R_6^0(\rho ;\varepsilon )$	Secondary spherical
23		2	$\scriptsize{\frac{{\left[ \begin{array}{l} 15(1 + 4{\varepsilon ^2} + 10{\varepsilon ^4} + 4{\varepsilon ^6} + {\varepsilon ^8}){\rho ^6} - 20(1 + 4{\varepsilon ^2} + 10{\varepsilon ^4} + 10{\varepsilon ^6} + 4{\varepsilon ^8} \\ + {\varepsilon ^{10}}){\rho ^4} + 6(1 + 4{\varepsilon ^2} + 10{\varepsilon ^4} + 20{\varepsilon ^6} + 10{\varepsilon ^8} + 4{\varepsilon ^{10}} + {\varepsilon ^{12}}){\rho ^2} \\ \end{array} \right]}}{{\left\{ {{{(1 - {\varepsilon ^2})}^2}{{\left[ \begin{array}{l} (1 + 4{\varepsilon ^2} + 10{\varepsilon ^4} + 4{\varepsilon ^6} + {\varepsilon ^8})(1 + 9{\varepsilon ^2} + 45{\varepsilon ^4} \\ + 65{\varepsilon ^{\text{6}}} + 45{\varepsilon ^8} + 9{\varepsilon ^{10}} + {\varepsilon ^{12}}) \\ \end{array} \right]}^{1/2}}} \right\}}}}$	$\sqrt {14} R_6^2(\rho ;\varepsilon )\sin 2\theta$	Tertiary Y astigmatism
24		2		$\sqrt {14} R_6^2(\rho ;\varepsilon )\cos 2\theta$	Tertiary X astigmatism
25		4	$\frac{{6{\rho ^6} - 5[(1 - {\varepsilon ^{12}})/(1 - {\varepsilon ^{10}})]{\rho ^4}}}{{{{\left\{ {{{(1 - {\varepsilon ^2})}^{ - 1}}[36(1 - {\varepsilon ^{14}}) - 35{{(1 - {\varepsilon ^{12}})}^2}/(1 - {\varepsilon ^{10}})]} \right\}}^{1/2}}}}$	$\sqrt {14} R_6^4(\rho ;\varepsilon )\sin 4\theta$
26		4		$\sqrt {14} R_6^4(\rho ;\varepsilon )\cos 6\theta$
27		6	$\frac{{{\rho ^6}}}{{{{(1 + {\varepsilon ^2} + {\varepsilon ^4} + {\varepsilon ^6} + {\varepsilon ^8} + {\varepsilon ^{10}} + {\varepsilon ^{12}})}^{1/2}}}}$	$\sqrt {14} R_6^6(\rho ;\varepsilon )\sin 4\theta$
28		6		$\sqrt {14} R_6^6(\rho ;\varepsilon )\cos 6\theta$

The non-normalized Zernike annular polynomials can be obtained by dropping the normalization factors from the orthonormal Zernike annular polynomials as:

$$\begin{equation}{Z_j}(\rho ,\theta ;\varepsilon ) = Z_n^m(\rho ,\theta ;\varepsilon ) = \left\{\! {\begin{array}{*{20}{l}} {R_n^m(\rho ;\varepsilon )\cos m\theta ,}\,{m \ne 0,\,\, j {\text{ is even}},} \\ {R_n^m(\rho ;\varepsilon )\sin m\theta ,}\;{m \ne 0, \,\,j {\text{ is odd,}}} \\ {R_n^0(\rho ;\varepsilon ),}\;\qquad\;\;\;{m = 0.} \end{array} } \right.\end{equation} \tag{ 98 }$$

They satisfy the following orthogonality condition:

$$\begin{equation}\frac{{\int_0^{2\pi } {\int_\varepsilon ^1 {{Z_j}(\rho ,\theta ;\varepsilon )} } {Z_{{j}{^{\prime}}}}(\rho ,\theta ;\varepsilon )\rho {\text{d}}\rho {\text{d}}\theta }}{{\int_0^{2\pi } {\int_\varepsilon ^1 {\rho {\text{d}}\rho {\text{d}}\theta } } }} = \frac{{1 + {\delta _{m0}}}}{{2(n + 1)}}{\delta _{jj^{\prime}}}.\end{equation} \tag{ 99 }$$

Figures 25 and 26 show the 3D visualization of the non-normalized Zernike annular polynomials up to the sixth degree for = 0.6 and their corresponding interferometric fringe patterns as in optical testing [37].

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3.2.1.2. Complex Zernike annular polynomials

Complex Zernike annular polynomials have normalized and non-normalized forms. The normalized complex Zernike annular polynomials defined in the Noll indices can be written as [4, 35]:

$$\begin{equation}V\,_n^l(\rho ,\theta ;\varepsilon ) = \sqrt {n + 1} R_n^l(\rho ;\varepsilon )\exp ({\text{i}}l\theta ),\end{equation} \tag{ 100 }$$

where n is a non-negative integer, l is an integer, n − |l| ⩾ 0 and is even, and the radial polynomial is defined as:

$$\begin{align} R_n^l(\rho ;\varepsilon )& = \omega _n^{|l|} \Bigg\{ {R_n^{|l|}(\rho ) } - \sum\limits_{i = 1}^{(n - |l|)/2} (n - 2i + 1)\nonumber\\ & \quad\times \overline {R_n^{|l|}(\rho )R_{n - 2i}^{|l|}(\rho ;\varepsilon )} R_{n - 2i}^{|l|}(\rho ;\varepsilon ) \Bigg\}, \end{align} \tag{ 101 }$$

and the weighting factor $\omega _n^{|l|}$ is given by:

$$\begin{equation}\omega _n^{|l|} = {\left( {\frac{{\dfrac{{1 - {\varepsilon ^2}}}{{2(n + 1)}}}}{{\int_\varepsilon ^1 {{{\left\{ {R_n^{|l|}(\rho ) - \sum\limits_{i = 1}^{(n - |l|)/2} {(n - 2i + 1)R_{n - 2i}^{|l|}(\rho ;\varepsilon )\overline {R_n^{|l|}(\rho )R_{n - 2i}^{|l|}(\rho ;\varepsilon )} } } \right\}}^2}\rho {\text{d}}\rho } }}} \right)^{1/2}}.\end{equation} \tag{ 102 }$$

The normalized complex Zernike annular polynomials satisfy the following orthonormality condition:

$$\begin{equation}\frac{{\int_0^{2{{\pi }}} {\int_\varepsilon ^1 {{{[V\,_n^l(\rho ,\theta ;\varepsilon )]}^ * }} } V_{{n}{^{\prime}}}^{{l}{^{\prime}}}(\rho ,\theta ;\varepsilon )\rho {\text{d}}\rho {\text{d}}\theta }}{{\int_0^{2{{\pi }}} {\int_\varepsilon ^1 {\rho {\text{d}}\rho {\text{d}}\theta } } }} = {\delta _{{ll}^{^{\prime}}}}{\delta _{{nn}{^{\prime}}}}.\end{equation} \tag{ 103 }$$

The non-normalized complex Zernike annular polynomials can be written as:

$$\begin{equation}V\,_n^l(\rho ,\theta ;\varepsilon ) = R_n^l(\rho ;\varepsilon )\exp ({\text{i}}l\theta ).\end{equation} \tag{ 104 }$$

They satisfy the following orthogonality condition:

$$\begin{equation}\frac{{\int_0^{2{{\pi }}} {\int_\varepsilon ^1 {{{[V\,_n^l(\rho ,\theta ;\varepsilon )]}^ * }} } V_{{n}{^{\prime}}}^{{l}{^{\prime}}}(\rho ,\theta ;\varepsilon )\rho {\text{d}}\rho {\text{d}}\theta }}{{\int_0^{2{{\pi }}} {\int_\varepsilon ^1 {\rho {\text{d}}\rho {\text{d}}\theta } } }} = \frac{1}{{n + 1}}{\delta _{{ll}^{^{\prime}}}}{\delta _{{nn}{^{\prime}}}}.\end{equation} \tag{ 105 }$$

3.2.1.3. Summary.

The definitions for the real and complex Zernike annular polynomials are summarized in table 17.

Table 17. Summary of the definition of Zernike annular polynomials.

	Real Zernike annular polynomials	Complex Zernike annular polynomials
Sources	Tatian, 1974 [5] Mahajan, 1981 [4] Zemax [25]	Mahajan, 1981 [4]
Indexing	The Noll indices	The Noll indices
Definition	$\begin{array}{l} {Z_j}(\rho ,\theta ;\varepsilon ) = Z_n^m(\rho ,\theta ;\varepsilon ) \\[4pt] = N_n^m\left\{ {\begin{array}{*{20}{l}} {R_n^m(\rho ;\varepsilon )\cos m\theta ,}\;{m \ne 0, \;j {\text{ is even}}} \\[4pt] {R_n^m(\rho ;\varepsilon )\sin m\theta ,}\;\;{m \ne 0,\; j {\text{ is odd}}} \\[4pt] {R_n^m(\rho ;\varepsilon ),}\;\qquad \quad {m = 0} \end{array} } \right. \\ \end{array}$	$V\,_n^l(\rho ,\theta ;\varepsilon ) = N_n^lR_n^l(\rho ;\varepsilon )\exp ({\text{i}}l\theta )$
Radial polynomials	Equation (94)	Equation (101)
Coordinate system	0 ⩽ ρ ⩽ 1 θ from +x axis anticlockwise	0 ⩽ ρ ⩽ 1 θ from +x axis anticlockwise
Normalization	Optional	Optional
Normalization factor	$N_n^m = \left\{ \begin{array}{l} 1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{non-normalized}} \\[6pt] \sqrt {\frac{{2(n + 1)}}{{(1 + {\delta _{m0}})}}} ,\;{\text{normalized}} \\ \end{array} \right.$	$N_n^l = \left\{ \begin{array}{l} 1,\;\;\;\;\;\;\;\;{\text{non-normalized}} \\[6pt] \sqrt {n + 1} ,\;{\text{normalized}} \\ \end{array} \right.$
Term number	Infinite	Infinite
Indices	j = 1, 2, 3, ..., ∞ n: non-negative integer m: non-negative integer n − m ⩾ 0 and even	n: non-negative integer l: integer n − |l| ⩾ 0 and even
Relationship between indices	$j = \left\{ {\begin{array}{*{20}{l}} {\frac{{n(n + 1)}}{2} + 1,}&{m = 0} \\[6pt] {\left[ {\frac{{n(n + 1)}}{2} + m,\frac{{n(n + 1)}}{2} + m + 1} \right],}&{m \ne 0} \end{array}} \right.$ $\begin{array}{l} n = \left\lfloor {\left( {\sqrt {2j - 1} + 0.5} \right) - 1} \right\rfloor \\[8pt] m = \left\{ {\begin{array}{*{20}{l}} {2 \times \left\lfloor {\frac{{2j + 1 - n\left( {n + 1} \right)}}{4}} \right\rfloor ,}&{n{\text{ is even}}} \\[8pt] {2 \times \left\lfloor {\frac{{2\left( {j + 1} \right) - n\left( {n + 1} \right)}}{4}} \right\rfloor - 1,}&{n{\text{ is odd}}} \end{array}} \right. \\ \end{array}$	—
Ordering	n, m both in ascending order	—

3.2.2.1. Orthogonality.

The orthogonal relationships of real and complex Zernike annular polynomials have been presented and can be found in equations (97), (99), (103) and (105). Moreover, the radial polynomials of Zernike annular polynomials are also orthogonal over the annular aperture and satisfy the following relationships [4]:

$$\begin{equation}\int_\varepsilon ^1 {R_n^m(\rho ;\varepsilon )R_{{n}{^{\prime}}}^m(\rho ;\varepsilon )\rho {\text{d}}\rho = \frac{{1 - {\varepsilon ^2}}}{{2(n + 1)}}{\delta _{{nn}{^{\prime}}}}} .\end{equation} \tag{ 106 }$$

3.2.2.2. Recurrence relation.

The recurrence relationship for generating radial polynomials of Zernike annular polynomials was derived by Tatian in 1974 [4, 5] and can be written as:

$$\begin{align}& R_n^m(\rho ;\varepsilon )\nonumber\\ & \quad = \left\{ \begin{gathered} R_{n = 2k}^0(\rho ;\varepsilon ) = R_{2k}^0\left[ {{{\left( {\frac{{{\rho ^2} - {\varepsilon ^2}}}{{1 - {\varepsilon ^2}}}} \right)}^{1/2}}} \right],\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;m = 0{\text{;}} \\ R_{n = 2l + m}^m(\rho ;\varepsilon ) = {\left[ {\frac{{1 - {\varepsilon ^2}}}{{2(2l + m + 1)h_l^m}}} \right]^{1/2}}{\rho ^m}Q_l^m(u),\;\;m \ne 0, \\ \end{gathered} \right.\end{align} \tag{ 107 }$$

where n and m obey the same conditions defined in Zernike annular polynomials (n and m are non-negative integers, n − m⩾ 0 and is even); k is a non-negative integer (k = 0, 1, 2, ..., ∞); l = (n − m)/2; u = ρ²; $Q_l^m(u)$ is a set of orthogonal polynomials obtained by orthogonalizing the sequence 1, u, ..., u^l over the interval (², 1) with a weight function u^m and can be written as [4, 5]:

$$\begin{equation}Q_l^m(u) = \left\{ \begin{aligned} & R_{2l}^0\left[ {{{\left( {\frac{{{\rho ^2} - {\varepsilon ^2}}}{{1 - {\varepsilon ^2}}}} \right)}^{1/2}}} \right],\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\quad\quad\;\; m = 0; \\ & \!\!\frac{{2(2l + 2m - 1)}}{{(l + m)(1 - {\varepsilon ^2})}}\frac{{h_l^{m - 1}}}{{Q_l^{m - 1}(0)}}\sum\limits_{i = 0}^l {\frac{{Q_i^{m - 1}(0)Q_i^{m - 1}(u)}}{{h_i^{m - 1}}},\;m \ne 0.} \\ \end{aligned} \right.\end{equation} \tag{ 108 }$$

The coefficient $h_l^m$ is:

$$\begin{equation}h_l^m = \left\{ \begin{gathered} \!\!\! \frac{{1 - {\varepsilon ^2}}}{{2(2l + 1)}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\quad\quad\;\; m = 0; \\ - \frac{{2(2l + 2m - 1)}}{{(l + m)(1 - {\varepsilon ^2})}}\frac{{Q_{l + 1}^{m - 1}(0)}}{{Q_l^{m - 1}(0)}}h_l^{m - 1},\;\;\;m \ne 0. \\ \end{gathered} \right.\end{equation} \tag{ 109 }$$

Especially, when m = n,

$$\begin{equation}R_n^n(\rho ;\varepsilon ) = \frac{{{\rho ^n}}}{{{{\left( {\sum\limits_{i = 0}^n {{\varepsilon ^{2i}}} } \right)}^{1/2}}}}.\end{equation} \tag{ 110 }$$

The above recurrence relationship can be initialized with $R_0^0(\rho ;\varepsilon ) = 1$.

3.2.2.3. The Fourier transform.

The Fourier transform of Zernike annular polynomials is derived by Dai and Mahajan [95] and can be written as:

$$\begin{align} {\mathcal{Z}_j}(r,\phi ;\varepsilon ) & = \int_0^{2{{\pi }}} {\int_\varepsilon ^1 {{Z_j}} } (\rho ,\theta ;\varepsilon )\nonumber\\[6pt] & \quad \times \exp \left[ { - {\text{i}}2{{\pi }}r\rho \cos (\theta - \phi )} \right]\rho {\text{d}}\rho {\text{d}}\theta \nonumber\\[6pt] & = \left\{ {\begin{array}{*{20}{l}} {\sqrt {2(n + 1)} H_n^m(r;\varepsilon )\cos (m\phi ),}&{m \ne 0,\;j {\text{ is }} {\text{even}},} \\ {\sqrt {2(n + 1)} H_n^m(r;\varepsilon )\sin (m\phi ),}&{m \ne 0,\;j {\text{ is }} {\text{odd}},} \\ {\sqrt {(n + 1)} H_n^m(r;\varepsilon ),}&{m = 0,} \end{array}} \right. \end{align} \tag{ 111 }$$

where (r, φ) denotes the polar coordinates in the frequency domain and:

$$\begin{align} & H_n^m(r;\varepsilon )\nonumber\\& \quad = \frac{1}{r}\sum\limits_{{n}{^{\prime}} = 0}^n {\Bigg\{ {{g_{{n}{^{\prime}}}}(n,m;\varepsilon ){{( - 1)}^{(n^{\prime}/2) + m}}} } \nonumber\\ & \qquad { \times \left[\! {{J_{{n}{^{\prime}} + 1}}(2\pi r) - \varepsilon \sum\limits_{{n}^{^{\prime\prime}} = 0}^{{n}^{^{\prime}}}\! {{{( - 1)}^{(n^{^{\prime\prime}} - {n}^{^{\prime}})/2}}{h_{{n}^{^{\prime\prime}}}}(n^{\prime};\varepsilon ){J_{{n}^{^{\prime\prime}} + 1}}(2{{\pi }}\varepsilon r)} }\! \right]}\! \Bigg\}, \end{align} \tag{ 112 }$$

where J is the Bessel function of the first kind (equation (82)) and:

$$\begin{equation}{g_{{n}{^{\prime}}}}(n,m;\varepsilon ) = \omega _n^m(\varepsilon )\frac{2}{{1 - {\varepsilon ^2}}}\int_\varepsilon ^1 {R_n^m(\rho )} R_{{n}{^{\prime}}}^m(\rho ;\varepsilon )\rho {\text{d}}\rho ,\end{equation} \tag{ 113 }$$

$$\begin{align}& {h_{{n}^{^{\prime\prime}}}}(n;\varepsilon ) \nonumber\\ & \quad = \sum\limits_{s = 0}^{(n - m)/2} {\sum\limits_{s^{\prime} = 0}^{[(n - m)/2] - s} {\frac{{{{( - 1)}^s}{\varepsilon ^{n - 2s}}(n - s)!(n + 1 - 2s - 2s^{\prime})}}{{s!s^{\prime}!(n + 1 - 2s - s^{\prime})!}}} } .\end{align} \tag{ 114 }$$

A list of the first few terms for $H_n^m$, ${g_{{n}{^{\prime}}}}$, and ${h_{{n}^{^{\prime\prime}}}}$ can be found in [95]. The expression in equation (111) reduces to the Fourier transform of Zernike circle polynomials (equation (30)) when = 0.

The orthogonality of Zernike annular polynomials makes them an excellent basis for wavefront analysis in annular optical systems. An annular wavefront can be represented by the linear combination of finite terms of Zernike annular polynomials as [113]:

$$\begin{equation}W(\rho ,\theta ;\varepsilon ) = \sum\limits_{j = 1}^J {{a_j}{Z_j}(\rho ,\theta ;\varepsilon ),} \end{equation} \tag{ 115 }$$

where J is the total terms of the polynomials, a_j is the expansion coefficients, and Z_j is the jth-term Zernike annular polynomial. The equation can be equivalently expressed in Cartesian coordinates as:

$$\begin{equation}W(x,y;\varepsilon ) = \sum\limits_{j = 0}^J {{a_j}{Z_j}(x,y;\varepsilon )} .\end{equation} \tag{ 116 }$$

Written in discrete and matrix forms, equation (51) becomes:

$$\begin{equation}{\mathbf{Aa = W}},\end{equation} \tag{ 117 }$$

where:

$$\begin{align} {\mathbf{A}} &= \left[ {\begin{array}{*{20}{c}} {{Z_1}({x_1},{y_1};\varepsilon )}& {{Z_2}({x_1},{y_1};\varepsilon )}& \cdots & {{Z_J}({x_1},{y_1};\varepsilon )} \\[4pt] {{Z_1}({x_2},{y_2};\varepsilon )}& {{Z_2}({x_2},{y_2};\varepsilon )} & \cdots & {{Z_J}({x_2},{y_2};\varepsilon )} \\[4pt] \vdots & \vdots & \ddots & \vdots \\[4pt] {{Z_1}({x_K},{y_K};\varepsilon )}& {{Z_2}({x_K},{y_K};\varepsilon )} & \cdots & {{Z_J}({x_K},{y_K};\varepsilon )} \end{array}} \right], \nonumber\\ {\mathbf{a}} &= \left[ {\begin{array}{*{20}{c}} {{a_1}} \\[4pt] {{a_2}} \\[4pt] \vdots \\[4pt] {{a_J}} \end{array}} \right], {\mathbf{W}} = \left[ {\begin{array}{*{20}{c}} {W({x_1},{y_1};\varepsilon )} \\[4pt] {W({x_2},{y_2};\varepsilon )} \\[4pt] \vdots \\[4pt] {W({x_K},{y_K};\varepsilon )} \end{array}} \right], \end{align} \tag{ 118 }$$

where K is the total number of data points within the unit circle. Generally, equation (117) is an overdetermined linear system, where there are more equations (K) than unknowns (J). It can be written into the normal equation [34, 68]:

$$\begin{equation}{{\mathbf{A}}^{\text{T}}}{\mathbf{A}}{\mathbf{a}} = {{\mathbf{A}}^{\text{T}}}{\mathbf{W}}{\text{,}}\end{equation} \tag{ 119 }$$

where the superscript T denotes matrix transpose. The solution can be obtained by matrix inversion as:

$$\begin{equation}{\mathbf{a}} = {({{\mathbf{A}}^{\text{T}}}{\mathbf{A}})^{ - 1}}{{\mathbf{A}}^{\text{T}}}{\mathbf{W}}.\end{equation} \tag{ 120 }$$

Figure 27 shows an example illustrating annular wavefront decomposition using 28-term orthonormal Zernike annular polynomials under the Noll indices. The amplitude of each coefficient indicates the strength of corresponding aberrations (table 16).

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Zernike polynomials have important applications in the diffraction theory of aberrations, which is concerned with the study of how wavefront aberrations affect image formation in practical optical systems [32, 47]. In a perfect optical imaging system, the light waves from a point object emerge in the image space as spherically convergent waves and form the well-known Airy pattern. However, a perfect imaging system never exists in practice. Waves emerging from a practical optical system deviate from a spherical wave and possess complicated forms.

Consider the wave propagation model illustrated in figure 29, where an aberrated wavefront at the exit pupil converges to the image plane. Let W_a and W_r denote the aberrated wavefront and its Gaussian reference wavefront in the unit of length, respectively. The position of the exit pupil is defined by the Cartesian coordinates (x, y, z) or the cylindrical coordinates (ρ, θ, z); the position of the image plane is defined by the Cartesian coordinates (ξ, η, ζ) or the cylindrical coordinates (r, φ, υ). The complex amplitude distribution at the exit pupil, called the pupil function, can be written as [114]:

$$\begin{equation}P(\rho ,\theta ) = A(\rho ,\theta )\exp \left[ {{\text{i}}\Phi (\rho ,\theta )} \right],\end{equation} \tag{ 121 }$$

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where A(ρ, θ) is the amplitude function and Φ is the phase function in the form of:

$$\begin{equation}\Phi (\rho ,\theta ) = \frac{{{\text{2}}\pi }}{\lambda }({W_a} - {W_r}),\end{equation} \tag{ 122 }$$

where λ is the wavelength. According to the scalar Debye integral [32, 114], the normalized complex amplitude, U(r, φ, υ), in the focal region of the image plane is given by:

$$\begin{align} U(r,\phi ,\upsilon ) & = \frac{1}{{{\pi }}}\iint {\exp ({\text{i}}\upsilon {\rho ^2})P(\rho ,\theta )} \nonumber\\ & \quad \times \exp \left[ {{\text{i}}2{{\pi }}r\rho \cos (\theta - \phi )} \right]\rho {\text{d}}\rho {\text{d}}\theta \nonumber\\ & = \frac{1}{{{\pi }}}\iint {\exp ({\text{i}}\upsilon {\rho ^2})A(\rho ,\theta )\exp \left[ {{\text{i}}\Phi (\rho ,\theta )} \right]} \nonumber\\ & \quad \times \exp \left[ {{\text{i}}2{{\pi }}r\rho \cos (\theta - \phi )} \right]\rho {\text{d}}\rho {\text{d}}\theta , \end{align} \tag{ 123 }$$

where υ is defined as the negative axial coordinate (-ζ) normalized with respect to the axial diffraction unit, $\lambda /(\pi s_0^2)$, and s₀ is the numerical aperture (NA) of the focusing beam. When the image plane is at the best focus (υ = 0), the complex amplitude U(r, φ, υ) in equation (123) reduces to the Fourier transform of the pupil function P(ρ, θ).

The PSF, defined as the diffraction pattern of a point object in the image plane, can be written as the squared modulus of the complex amplitude U [115, 116], i.e.

$$\begin{equation}{\text{PSF}} = {\left| {U(r,\phi ,\upsilon )} \right|^2}.\end{equation} \tag{ 124 }$$

The image of an extended object formed by an optical system is the convolution of the object itself with the PSF of the system, which can be mathematically modeled as [117]:

$$\begin{equation}g = f * {\text{PSF}},\end{equation} \tag{ 125 }$$

where f and g denote the object and the image, respectively, and ∗ represents convolution. To understand the impact of wavefront aberrations on the final image quality, the PSF needs to be evaluated. In the next section, we briefly review the analytical PSF computation approaches first developed by Nijboer and Zernike [6, 21] and later extended by Janssen [8], where expanding wavefront aberrations at the exit pupil using Zernike circle polynomials is the key.

In general, analytical evaluation of the diffraction integrals in equation (123) is difficult except for some specific cases. In 1942, Bernard Nijboer, a PhD, student of Zernike, expanded the aberration function at the exit pupil into a series of Zernike circle polynomials and formulated an efficient representation of the complex amplitude distribution in the image plane [6, 21]. This work allows analytical evaluation of the diffraction integral and the PSF of a general optical system and is referred to as the Nijboer–Zernike theory. For completeness, we briefly review the basic principle of the theory, which is well summarized in [114].

In the Nijboer–Zernike theory, the pupil function is assumed to be uniform in amplitude and thus can be written as a purely phase-aberrated function, i.e.

$$\begin{equation}P(\rho ,\theta ) = \exp \left[ {{\text{i}}\Phi (\rho ,\theta )} \right].\end{equation} \tag{ 126 }$$

Expanding the pupil function into the Taylor series, the diffraction integral in equation (123) becomes:

$$\begin{align} U(r,\phi ,\upsilon ) &= \frac{1}{{{\pi }}}\sum\limits_{k = 0}^\infty {\frac{{{i^k}}}{{k!}}\int_0^{2\pi } {\int_0^1 {\exp ({\text{i}}\upsilon {\rho ^2}){\Phi ^k}(\rho ,\theta )} } } \nonumber \\ &\quad \times \exp \left[ {{\text{i}}2{{\pi }}r\rho \cos (\theta - \phi )} \right]\rho {\text{d}}\rho {\text{d}}\theta .\end{align} \tag{ 127 }$$

Expanding the phase function, Φ(ρ, θ), into a set of Zernike circle polynomials gives:

$$\begin{equation}\Phi (\rho ,\theta ) = \sum\limits_{n,m} {\alpha _n^mZ_n^m(\rho ,\theta )} .\end{equation} \tag{ 128 }$$

Substituting the Zernike expansion into the integral (equation (127)) and performing the integration over θ using elementary Bessel function operations, we obtain:

$$\begin{align} U(r,\phi ,\upsilon ) & \approx 2\int_0^1 {\exp ({\text{i}}\upsilon {\rho ^2}){J_0}(2{{\pi }}\rho r)\rho {\text{d}}\rho } \nonumber\\ & \quad + 2{\text{i}}\sum\limits_{n,m} {{\text{i}}^m}\alpha _n^m\cos (m\phi )\nonumber\\ & \quad \times \int_0^1 {\exp ({\text{i}}\upsilon {\rho ^2})} R_n^m(\rho ){J_m}(2{{\pi }}\rho r)\rho {\text{d}}\rho , \end{align} \tag{ 129 }$$

where $\alpha _n^m$ is the Zernike expansion coefficients and J_m is a Bessel function of the first kind and of order m. Note that in the reduction, the phase aberration is considered small enough so that truncation of the infinite series in equation (128) after the term k = 1 is allowed. The above equation can be further reduced using the relationship in equation (83) as:

$$\begin{align}U(r,\phi ,\upsilon = 0) = \frac{{{J_1}(2{{\pi }}r)}}{{{{\pi }}r}} + \sum\limits_{n,m}^{\prime} {{{\text{i}}^{n + 1}}\frac{{{J_{n + 1}}(2{{\pi }}r)}}{{2{{\pi }}r}}\left[ {\alpha _n^m\cos (m\phi )} \right]} ,\end{align} \tag{ 130 }$$

where the prime indicating that m = n = 0 should be excluded from the summation.

The expression of the complex amplitude, U, provides an analytical method for evaluating the PSF of an optical system. Although elegant in expression, the Nijboer–Zernike approach is not widely used in practice [114], largely because that the derivation requires the amplitude over the pupil to be uniform and the wavefront aberration is restricted to be sufficiently small (in the order of a fraction of the wavelength [32]).

Figure 30 illustrates the appearance of the PSF of an optical system when only a single Zernike term (root mean square value: 0.1 μm, wavelength: 570 nm) is present in the wavefront aberration. Figure 31 presents an example showing that wavefront aberrations degrade the image quality of an optical system.

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The Nijboer–Zernike theory is only valid in the case of small aberrations and can only produce accurate values of PSF at positions close to geometrical focus. To deal with the problem, Janssen in 2002 formulated a general expression in terms of the power-Bessel series and extended the Nijboer–Zernike theory for optical systems with large aberrations [8, 114, 118, 119]. The extended Nijboer–Zernike theory can analytically compute the PSF of an aberrated optical system described by Zernike coefficients and accelerates further developments in focused field diffraction theory.

The extended Nijboer–Zernike theory adopts a generalized definition for the pupil function and expands it using Zernike circle polynomials as [114]:

$$\begin{equation}P(\rho ,\theta ) = A(\rho ,\theta )\exp [{\text{i}}\Phi (\rho ,\theta )] = \sum\limits_{n,m} {\beta _n^mZ_n^m(\rho ,\theta )} ,\end{equation} \tag{ 131 }$$

where the amplitude, A(ρ, θ), and phase aberration, Φ(ρ, θ), are real-valued; the coefficients, $\beta _n^m$, is in general complex-valued; $Z_n^m$ is the non-normalized complex Zernike circle polynomials defined in equation (16). Substituting the pupil function into the diffraction integral (equation (123)), the complex amplitude, U, can be expressed as:

$$\begin{equation}U(r,\phi ,\upsilon ) = 2\sum\limits_{n,m} {\beta _n^m{{\text{i}}^{|m|}}K_n^{|m|}\exp ({\text{i}}m\phi ),} \end{equation} \tag{ 132 }$$

where:

$$\begin{equation}K_n^m = \int_0^1 {\exp ({\text{i}}\upsilon {\rho ^2})R_n^{|m|}(\rho ){J_m}(2{{\pi }}\rho r)\rho {\text{d}}\rho } .\end{equation} \tag{ 133 }$$

Janssen derived a Bessel series representation for the integral in equation (133) and reformulate the above equation as:

$$\begin{equation}K_n^m = {\varepsilon _m}\exp ({\text{i}}\upsilon )\sum\limits_{l = 1}^\infty {{{( - 2{\text{i}}\upsilon )}^{l - 1}}\sum\limits_{j = 0}^p {{v_{lj}}\frac{{{J_{|m| + l + 2j}}(v)}}{{l{v^l}}}} } ,\end{equation} \tag{ 134 }$$

where:

$$\begin{equation}{v_{lj}} = \frac{{{{( - 1)}^p}(|m| + l + 2j)\left( {\begin{array}{*{20}{c}} {|m| + j + l - 1} \\ {l - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {j + l - 1} \\ {l - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {l - 1} \\ {p - j} \end{array}} \right)}}{{\left( {\begin{array}{*{20}{c}} {q + l + j} \\ l \end{array}} \right)}},\end{equation} \tag{ 135 }$$

$$\begin{align}v & = 2\pi r,{\varepsilon _m} \nonumber\\[6pt] & = \left\{ {\begin{array}{*{20}{l}} { - 1,}&{{\text{odd }} m < 0,} \\ {1,}&{{\text{others,}}} \end{array}} \right.,\ p = \frac{{(n - |m|)}}{2},q = \frac{{(n + |m|)}}{2}.\end{align} \tag{ 136 }$$

The symbol $\left( \begin{gathered} n \\ k \\ \end{gathered} \right)$ denotes combination and is defined as:

$$\begin{equation}\left( \begin{gathered} n \\ k \\ \end{gathered} \right) = \frac{{n!}}{{k!(n - k)!}}.\end{equation} \tag{ 137 }$$

Note that the equation (134) suffers from loss-of-digits and slow convergence for larger υ under standard precision. An advanced version of the ENZ-theory has been developed to virtually eliminate the convergence problem by replacing the power-Bessel series in equation (134) with Bessel-Bessel series [120]. Using equation (134), we can compute the PSF of an optical system with an exit pupil defined by a set of β-coefficients (equation (131)). The extended Nijboer–Zernike theory has been used in several applications, such as aberration retrieval in high-NA optical lithography systems [121–123] and acoustic diffraction problems [124, 125].

Zernike moments for image analysis and pattern recognition were first introduced by Teague in 1980 [12]. They are defined over a unit circle by employing Zernike polynomials as the basis function and can be written as [12]:

$$\begin{equation}{M_{nl}} = \frac{{n + 1}}{{{\pi }}}\iint\limits_{{x^2} + {y^2} \leqslant 1} {f(x,y){{\left[ {V\,_n^l(x,y)} \right]}^*}{\text{d}}x{\text{d}}y},\end{equation} \tag{ 154 }$$

where $V\,_n^l$ is the non-normalized complex Zernike polynomials (equation (16)), the asterisk denotes complex conjugate, and M_nl is the Zernike moment of degree n with repetition l. n is a non- negative integer, l is an integer, n − |l| ⩾ 0 and is even. The completeness and orthogonality of $V\,_n^l$ allow for the representation of a square integrable image function, f(x, y), defined on a unit disk using Zernike polynomials as [181, 182]:

$$\begin{equation}f(x,y) = \frac{{n + 1}}{{{\pi }}}\sum\limits_n {\sum\limits_l {{M_{nl}}V\,_n^l(x,y)} } .\end{equation} \tag{ 155 }$$

The expression in equation (155) suggests that the image, f(x, y), can be theoretically reconstructed from its Zernike moments. However, the practical importance of this property is not that significant because moments are not a good tool for image compression in general [183]. For a digital image, equation (154) can be written in discrete form as:

$$\begin{equation}{M_{nl}} = \frac{{n + 1}}{{{\pi }}}\sum\limits_x {\sum\limits_y {f(x,y){{\left[ {V\,_n^l(x,y)} \right]}^*}} } ,\end{equation} \tag{ 156 }$$

where x² + y² ⩽ 1.

The fundamental feature of Zernike moments is their rotational invariance. If the image, f(x, y) is rotated by an angle, α, the Zernike moments of the rotated image is given by [184]:

$$\begin{equation}{M^\prime_{nl}} = {M_{nl}}\exp ( - {\text{i}}l\alpha ).\end{equation} \tag{ 157 }$$

Equation (157) indicates that rotating an image will introduce a phase shift to the Zernike moments but will not alter the magnitudes. This simple property leads to the conclusion that the magnitudes of the Zernike moments, $|{M_{nl}}|$, can be used as rotationally invariant features of the image function, f(x, y). Moreover, bearing the facts that ${M_{n, - l}} = {M_{nl}}^*,\left| {{M_{n, - l}}} \right| = \left| {{M_{nl}}} \right|$ in mind, one may only use $|{M_{nl}}|$ with l ⩾ 0 as Zernike feature descriptors, as shown in table 18. Based on the Zernike moments, two primary rotation invariants, I_{n 0} and I_nl , can be constructed as [12, 185, 186]:

$$\begin{align} {I_{n0}} & = {M_{n0}}, \nonumber\\ {I_{nl}} & = {\left| {{M_{nl}}} \right|^2}. \end{align} \tag{ 158 }$$

Table 18. List of Zernike moments from degree zero to nine.

Degree	Moments	No. of Moments
0	${M_{00}}$	1
1	${M_{11}}$	1
2	${M_{20}},{M_{22}}$	2
3	${M_{31}},{M_{33}}$	2
4	${M_{40}},{M_{42}},{M_{44}}$	3
5	${M_{51}},{M_{53}},{M_{55}}$	3
6	${M_{71}},{M_{73}},{M_{75}},{M_{77}}$	4
7	${M_{71}},{M_{73}},{M_{75}},{M_{77}}$	4
8	${M_{80}},{M_{82}},{M_{84}},{M_{86}},{M_{88}}$	5
9	${M_{91}},{M_{93}},{M_{95}},{M_{97}},{M_{99}}$	5

They are the most important and most frequently used Zernike shape descriptors.

The rotation invariance of Zernike moments is illustrated by a numerical experiment. Figure 43 shows a gray image of 512 × 512 pixels and its five rotated versions with rotation angles of 60°, 120°, 180°, 240°, 300°, respectively. Table 19 presents the magnitudes of the Zernike moments up to the third degree and their statistics (mean μ and standard deviation σ). The data show that the standard deviations of the Zernike moments are close to zero, suggesting that Zernike moments are excellent shape descriptors for object recognition. In this example, the reason for not obtaining exact rotation invariance is due to discretization errors, which have been discussed by several authors [182, 187].

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Table 19. Magnitudes of some Zernike moments for the rotated images in figure 43 and their statistics.

	$\left| {{M_{00}}} \right|$	$\left| {{M_{11}}} \right|$	$\left| {{M_{20}}} \right|$	$\left| {{M_{22}}} \right|$	$\left| {{M_{31}}} \right|$	$\left| {{M_{33}}} \right|$
0°	29 686.85	2295.99	4959.74	3543.28	3219.67	1585.55
60°	29 682.93	2296.33	4969.33	3543.65	3217.36	1584.38
120°	29 685.44	2296.22	4964.34	3548.26	3220.37	1586.41
180°	29 686.85	2295.99	4959.74	3543.28	3219.67	1585.55
240°	29 682.93	2296.33	4969.33	3543.65	3217.36	1584.38
300°	29 685.44	2296.22	4964.34	3548.26	3220.37	1586.41
μ	29 685.07	2296.18	4964.47	3545.06	3219.13	1585.47
σ	1.78	0.15	4.29	2.48	1.41	0.91
σ/μ%	0.006	0.007	0.09	0.07	0.04	0.06

Fast computation of Zernike moments is an important topic in Zernike moments based image analysis. To compute Zernike moments at lower degrees, one can directly use the explicit definition of Zernike polynomials. However, for Zernike moments at higher degrees, this method is not recommended because it involves the calculation of factorial functions present in the radial polynomial, which is computationally expensive [59]. In these cases, one can employ fast computation algorithms, such as the recursive methods based on the recurrence relationships of the radial polynomials (section 2.2.6) [59, 62, 64], among others [63, 188]. The recursive algorithms are reportedly more efficient and particularly suitable for fast calculation of Zernike moments.

As an important orthogonal moment, Zernike moments employ Zernike polynomials as the basis function and overcome the drawbacks of geometric moments. They have minimum information redundancy and can represent features in a more efficient, irredundant way. To date, Zernike moments have been widely used in many applications, such as pattern recognition [184, 189–191], multimedia watermarking [192, 193], and medical image analysis [194, 195].

Pseudo Zernike moments stem from the pseudo Zernike polynomials (described in section 2.5.5), which are also a set of polynomials orthogonal over a unit circle analogous to the conventional Zernike polynomials. Pseudo Zernike moments, denoted by ${\mathcal{M}_{nl}}$, are defined as [48, 186]:

$$\begin{equation}{\mathcal{M}_{nl}} = \frac{{n + 1}}{{{\pi }}}\int_0^{2{{\pi }}} {\int_0^1 {f(\rho ,\theta ){{\left( {\mathcal{V}_n^l} \right)}^*}\rho {\text{d}}\rho {\text{d}}\theta } } ,\end{equation} \tag{ 159 }$$

where $\mathcal{V}_n^l$ is the pseudo Zernike polynomials defined in equation (86), n is a nonnegative integer, l is an integer, and n − |l| ⩾ 0. Pseudo Zernike moments are analogous to conventional Zernike moments (equation (154)) and also hold the property of rotation invariance. However, they eliminate the constrain of n − |l| = even and thus have more moment invariants for the same degree n [Pseudo Zernike moments contain (n+ 1)² invariants while Zernike moments have (n + 1)(n + 2)/2]. It is shown that pseudo Zernike moments are less sensitive to image noise than conventional Zernike moments [48]. Pseudo Zernike moments also have fast computation algorithms [196, 197] and have been used in a range of image analysis and pattern recognition applications [198, 199].