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Appendix
In this work, we characterize the input beam phase profile / i( x i, y i) within the input pupil plane of the SRI wavefront sensor, where x i and y i are coordinates for the horizontal and vertical dimensions, respectively. The position of the ordered pair( x i, y i) is defined by its radial distance from the origin / i =( x i2 + y i 2) 1 / 2 and azimuthal angle / i = arctan( y i / x i), counterclockwise off the + x i-axis. The radial distance spans outward to three times the input beam’ s radius x, giving 0 q i 3x, andtheazimuthal angle spans 0 h < 2p. The input beam phase profile can then be expanded in terms of orthogonal Zernike polynomials, Z m n ðq i = ð3xÞ; h i Þ, as [ 39 ]
/ i ðx i; y i Þ¼ / i ðq i = ð3xÞ; h i Þ¼Z jmj n ðq i = ð3xÞ; h i Þ
¼ Um n Rjmj n ðq i = ð3xÞÞ cosðmh i Þ; m 0
U m n R jmj ðq i = ð3xÞÞsin ðjmjh i Þ; m < 0; n ðA: 1Þ where U m n is a normalization factor, the non-negative integer index n is the radial degree, the integer index m is the azimuthal frequency, and the difference between n and | m | is even and greater than or equal to zero. These two integers define Zernike polynomials according to [ 39 ]
R jmj ðq i = ð3xÞÞ n n� ¼ Xj mj = 2 ð�1Þ s ðn � sÞ! ðq s! ðn þ jmj = 2 � sÞ! ðn � jmj = 2 � sÞ! i = 3xÞ n�2s: ðA: 2Þ s¼0
Table A. 1 lists these two integer indices with their associated Noll mode order J, asusedinthisworkandelsewhere [ 18, 26 ], and OSA / ANSI mode order, as used elsewhere [ 40 ]). The table then lists the normalized Zernike polynomials with descriptors for the associated wavefront aberration and even / odd symmetry.