J. Eur. Opt. Society-Rapid Publ. 22, 31( 2026) 321
8
><
>:
z �f2lg 2j ¼ t f2lg
2j
z þf2lg 2j ¼ t f2lg
2j
z �f2lþ1g 2jþ1 ¼ t f2lþ1g
2jþ1
z þf2lþ1g 2jþ1 ¼ t f2lþ1g
2jþ1
"
T 2j�2
1 0
# " # " #
fk > 0g ½bf2pþ1g 2l b fqg j�l Šflgfkg 0 X fqg
j x �1 y 0 0 diagðw2j
Þ
0 B e� j�1 X j�1
"
T 2j�2
1 0
# " # " #
fk > 0g ½bf2pg 2l b fqg j�l Šflgfkg 0 X fqg
j
0 0 diagðw2j Þ
0 B eþ j�1
X j�1
"
T 2j�1
1 0
# " # " #: ðA: 15Þ
fk > 0g ½bf2pþ1g 2lþ1 b fqg
j�l Šflgfkg 0 X fqg
j y 0 0 diagðw2jþ1 Þ
0 B o� j�1 X j�1
"
T 2j�1
1 0
# " # " #
fk > 0g ½bf2pg 2lþ1 b fqg
j�l Šflgfkg 0 X fqg
j x 0 0 diagðw2jþ1 Þ
0 B oþ j�1
X j�1
|
2 b 0 j�l
0
6
.
4
.
|
b 1 j�l b 0 j�l
.
.
|
.
..
|
b j�l j�l b j�l�1 j�l
.
.
|
0 b j�l j�l
.
.
|
.
.
|
0
0
.
.
|
3
7
5
|
0 |
|
0 |
b 0 j�l |
b 1 j�l |
|
b j�l j�l |
|
ðlþ1Þðjþ1Þ
; ðA: 17Þ
which performs the multiplication between the azimuth polynomials and the radial polynomials for n = 2j of even order, þm ¼þð2lÞ part, just as b f2pþ1g
2l b fqg j�l for �m ¼�ð2lÞ part, while b f2pg
2lþ1 b fqg j�l and b f2pþ1g
2lþ1 b fqg j�l stand for n ¼ 2j þ 1ofodd order, þm ¼ð2l þ 1Þ and �m ¼�ð2l þ 1Þ parts, respectively.
Overall, equation( A. 14) and its matrix form( equation( A. 15)) show two cases of block-wise recurrence in complete Zernike calculations, which are irrelevant to the coordinates, since equation( A. 15) can be represented as follows:
8 z �f2lg 2j ¼ T 2j diagðw fkg
2j ÞB e� j
X j x �1 y
><
>: z þf2lg 2j z �f2lþ1g 2jþ1 z þf2lþ1g 2jþ1
¼ T 2j diagðw fkg
2j ÞB eþ j
X j
¼ T 2jþ1 diagðw fkg
2jþ1ÞB o� j
X j y
¼ T 2jþ1 diagðw fkg
2jþ1ÞB oþ j
X j x where we have 8
B e� 0
¼ 0; B eþ
0
¼ 1
B o� 0
¼ 1; B oþ
0
¼ 1 " #
B e� j
¼ B� 2j
0 0 B e� j�1
" # ><
B eþ j
¼ Bþ 2j
0 0 B eþ j�1
" #
B o� j
¼ B� 2jþ1
0 0 B o�
>: j�1
" #
B oþ j
¼ Bþ 2jþ1
0 0 B oþ j�1 h; B � 2j ¼ bf2pþ1g 2l
h; B þ 2j ¼ bf2pg 2l h; B � 2jþ1 ¼ bf2pþ1g 2lþ1
h; B þ 2jþ1 ¼ bf2pg 2lþ1 i flgfkg b fqg j�l
i flgfkg b fqg j�l i flgfkg b fqg j�l i flgfkg b fqg j�l ðA: 18Þ
;
ðA: 19Þ such that both T n¼2j, T n¼2jþ1, B e j¼n = 2, andBo j¼ðn�1Þ = 2 have their recursive submatrix decomposition and are fully supported by an n-row Pascal triangle.
A. 4 Block-wise Zernike calculation Since w 0 n
1 always holds, all formulas in equation( A. 18) have an
alternative pseudo-iterative form. For example, we have z þf2lg 2j ¼ t flg
2j B þ 2j
X fqg j þ T 2ðj�1Þ
fk > 0g
diag w2j
B eþ j�1 X j�1 ðA: 20Þ
where denotes the Hadamard product as element-wise multiplication between a vector and a matrix, e. g., we have
2 3 t 0 t 1 t flg B flgfkg ¼ 6.
7 4
5
|
2
B 0; 0
|
B 0; 1 |
|
B 0; j
3
|
|
B 1; 0
6
4
.
|
B 1; 1
.
|
.
..
|
B 1; j
.
7
5
|
B j; 0 |
B j; 1 |
|
B j; j |
2 t 0 B 0; 0 |
t 0 B 0; 1 |
|
t 0 B 0; j
3
|
|
t 1 B 1; 0
¼
6
4
.
.
|
t 1 B 1; 1
.
.
|
.
..
|
t 1 B 1; j
.
7
.
5
|
t j B j; 0 |
t j B j; 1 |
|
t j B j; j |
Equation( A. 2) is equivalent to
z þf2lg 2j
¼ Xj k¼0 w k 2j t j ðjþ1Þ1
ðjþ1Þðjþ1Þ
ðjþ1Þðjþ1Þ
t flg 2ðj�kÞ Bþ 2ðj�kÞ
X fqg; ðj�kÞ
: ðA: 21Þ
ðA: 22Þ
which offers us a direct block-wise transformation method for calculating the cos-relevant components of even-order( n ¼ 2j) Zernike basis functions in the Cartesian coordinate system. Such a transformation applies to all Zernike basis functions, as shown in equation( 10).