3.5.2. (cont.) Zernike aberration coefficients

TABLE 5: Zernike circle polynomials for selected balanced (best focus) aberrations.
COLUMN I; "primary" refers to the lower, or 4th order wavefront aberration form, with n+m=4, and "secondary" to the subsequent higher, or 6th order form, with n+m=6 (in terms of ray aberrations, 3rd and 5th order, respectively);
COLUMN II: integers specifying Zernike mode; the subscript n, or radial order, is the highest exponent of ρ in the radial variable (analog to the exponent of normalized pupil height factor in the standard aberration function); superscript m specifies angular frequency of the mode, with positive integers - such as those given in the table - indicating change in deviation with cosine of the angular coordinate θ, and negative integers indicating its change with sine function (numerically, m is analog to the radial power in image space for standard aberration functions);
COLUMN III: General (orthogonal) form of the Zernike circle polynomial, V(ρ)cos(mθ), is a product of its radial variable V(ρ) and angular variable cos(mθ);
COLUMN IV: Orthonormal Zernike polynomial, normalized with a factor N=[2(n+1)/(1+δm0)]1/2 that scales it to unit variance; in effect, the normalizing factor N scales the value of the polynomial to its RMS error level, which makes polynomial values for different aberrations directly comparable, as well as additive/subtractive; it is directly related to the manner in which zero mean splits the wavefront: as FIG. 32 illustrates, with balanced primary spherical aberration, the P-V wavefront error split relative to zero mean is proportional to (1+0.5), with the P-V/RMS ratio given by 1.5√5; for balanced secondary, the split is even, proportional to (1+1), with the P-V/RMS ratio given by 2√7 (FIG. 37C); for tilt, defocus, primary coma and astigmatism (last three, FIG.30-1), the wavefront
P-V split vs. zero mean is proportional to (1+1) and the P-V/RMS factor is given by 4, 2√3, 2√8 and 2√6, respectively  (δm0 in the normalization factor N is Kronecker delta of m and 0, thus δm0=0 for m≠0 and δm0=1 for m=0); 
COLUMN V: Zernike aberration term, or mode, Z (ρ,θ), a product of the orthonormal polynomial P' and Zernike expansion coefficient znm, the latter equaling the RMS wavefront error in units of wavelength (the aberration term P' is what is popularly called "Zernike coefficient" value given by raytrace programs);
COLUMN VI: Relative (in units of wavelength) RMS wavefront error ωz corresponding to the Zernike aberration term; it is called  Zernike expansion coefficient, and obtained directly from the Zernike aberration term Z as ωz=znm=Z/NP, where coordinates for the orthogonal polynomial P=P (ρ,θ) are set to ρ=1 and θ=0 (thus giving 1 for any orthogonal polynomial P), resulting in ωz=znm=Z/N; denoting Z as a general Zernike aberration term , and terms with subscripts t, d, s1, s2, c1, c2, a1 and a2 as specifically referring to tilt, defocus, primary and secondary spherical, coma and astigmatism, respectively, the RMS error is obtained directly from the term value as indicated (obviously, any specific RMS error can be directly assigned the corresponding Zernike aberration term as Z=Nωz; for instance, for the diffraction limited RMS ω=1/√180, the corresponding Zernike term value for balanced lower-order spherical aberration is Zs1=Z=√5ω=√5z40=1/6).

ABERRATION	N	|Z|=N|z| (|z|=0.0745)	WFE
			Max. deviation from zero mean	P-V
Defocus	√3	0.1291	0.1291	0.2582
Primary astigmatism	√6	0.1826	0.1826	0.3652
Primary coma	√8	0.2108	0.2108	0.4216
Balanced primary spherical	√5	0.1666	0.1666	0.2500
Trefoil	√8	0.2108	0.2108	0.4216
Secondary astigmatism	√10	0.2357	0.2357	0.4714
Secondary coma	√12	0.2582	0.2582	0.5164
Secondary spherical	√7	0.1972	0.1972	0.3944

Value of the Zernike term reflects the maximum (along the axis of aberration) edge deviation from zero mean, also in units of wavelength. This maximum deviation is identical on both sides of zero mean (i.e. for the two opposite end points along the axis of aberration where the deviation, for ρ=1 and unit polynomial value is determined by cosθ or cos2θ - for the cosine term - thus equal in magnitude and opposite in sign for the extreme values of the trigonometric function) for all aberrations except primary spherical, where the maximum deviation on the opposite side of zero mean is half as large (as shown on FIG. 32; it can be found by determining the value of ρ for the maximum deviation on the opposite side of zero mean, for which the first derivative of 6ρ4-6ρ2+1 is zero). Thus the P-V error is 1.5N|z| for spherical aberration, and 2N|z| for all others.

These values also can be expressed in microns. For example, assuming 0.55μm wavelength, the corresponding values for defocus would have been |Z|=0.071μm for the term, and |z|=0.041μm for the coefficient. Obviously, Zernikes' value expressed in units of wavelength vary with the wavelength.

The combined P-V wavefront deviation for given set of Zernikes is a sum of the individual deviations for each term at every point of the wavefront. For instance, for the edge point above (ρ=1), it is given by the sum of all P-V values in the last column, with their sign being determined by that of the corresponding Zernike term. Combined RMS wavefront error for two or more Zernike terms is a square root of the sum of all the RMS wavefront errors individually squared.