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3.5.2. Zernike aberrations
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3.5.3. Zernike expansion schemes
► 3.5.2. (cont.) Zernike aberration coefficients
PAGE HIGHLIGHTS Since both, standard aberration functions and Zernike aberration polynomials, are describing the same wavefront deviations, they can be related and converted from one form to the other. Denoting Zernike aberration terms  the orthonormal Zernike polynomials  simply as Za (full designation , usually written as or, in a singleindex notation, as Zj, informally  and incorrectly  referred to as "Zernike coefficients"), and the corresponding Zernike expansion coefficients as za (usually written as cnm, here znm), where the term subscript a identifies the corresponding aberration as balanced primary spherical (S), coma (C) or astigmatism (A), Zernike polynomial form for these three pointimage quality primary aberrations can be related to the peak aberration coefficients S, C and A from Eq. 5.1 as follows:
 spherical aberration:
ZS
=
zS√5(6ρ46ρ2+1)
= S(6ρ46ρ2+1)/6,
with n=4, m=0, thus
ZS=Z
and
zS=z40 with "" indicating absolute  i.e. w/o numerical sign  value (note that the sign of m superscript and corresponding cosine function are according to the angle convention used on this site) This implies that Zernike expansion coefficients zS, zC and zA, equal the corresponding RMS wavefront error ω, which is in terms of the peak aberration coefficient given by ωS=S/6√5=S/√180, ωC=C/3√8=C/√72 and ωA=A/√24 for spherical aberration, coma and astigmatism, respectively. With respect to the Zernike aberration term, ωS=zS=ZS/√5, ωC=zC=ZC/√8 and ωA=zA=ZA/√6 (coma and astigmatism along the ais of aberration, i.e. for θ=0). Note that the RMS wavefront error is by definition numerically positive, unlike peak/PV wavefront error, or Zernike expansion coefficients, which can be numerically negative. Similarly, conversion for defocus is ZD= zD√3(2ρ21)= Pρ2/2, with n=2, m=0, thus ZD=Z and zD=z20 implying ωD=zD=ZD/2√3. Note that these relations are for best focus location; also, in order for the nominal error to reflect its actual effect on diffraction intensity distribution, expressing the expansion coefficients as representing the RMS wavefront error requires the latter to be nearly identical to the phase factor φ of standard deviation, i.e. phase error averaged over the pupil (requirement fulfilled for lowlevel aberrations affecting most or all of wavefront area, roughly below λ/2 PV in magnitude). The above relations are valid for clear aperture (Zernike circle polynomials/coefficients). To an aperture with central obstruction applies different polynomial form (Zernike annular polynomials/coefficients). In this case, all three  RMS wavefront error, Zernike expansion coefficient and Zernike aberration term change according to a factor appropriate to each aberration form. Specifically, ωSo= zSo = ZSo/√5 = ωS(1o2)2 = zS(1o2)2= ZS(1o2)2/√5 ωCo= zCo = ZCo/√8 = ZC(1o2)(1+4o2+o4)1/2/√8(1+o2)1/2 and ωAo= zAo= ZAo/√6 = ZA(1+o2+o4)1/2/√6 for primary spherical aberration, coma and astigmatism, respectively, with o being the relative obstruction size in units of the aperture. Following table gives an overview of the Zernike aberration forms for the most common monochromatic aberrations in telescopes, for clear circular aperture (aberrations in aperture with central obstruction are described with Zernike annular polynomials). The three pointimage aberrations, spherical, coma and astigmatism, are balanced, with "balanced" as before, referring to the principal aberration form that combines two or more secondary aberrations in order to reduce error to a minimum (i.e. to the level at its diffraction, or best focus).
For instance, balanced primary
spherical includes its principal aberration term
ρ4
and balancing defocus term ρ2,
coma includes its principal aberration term
ρ3
and balancing tilt term ρ,
secondary spherical, also in its balanced form (minimized by combining
it with 4th order spherical and defocus, thus here referred to as balanced
6th/4th order spherical aberration, in order to distinguish it from
balanced pure 6th order aberration, which is minimized by combining it with defocus
alone) includes its
principal aberration term ρ6
and two balancing terms, for lowerorder spherical and defocus (ρ4
and ρ2,
respectively). The polynomial forms are as given by Mahajan (Optical
Imaging and Aberrations).
TABLE 5: Zernike circle polynomials for
selected balanced (best focus) aberrations. Each separate polynomial in the above table describes single aberration of a perfect conical surface, hence only a single polynomial suffices to describe it; since the aberrations are separated, the wavefront orientation is inconsequential for describing the mode of deformation, and all radially nonsymmetrical aberrations are given with a positive m integer (i.e. with cosine function in the angular variable). As mentioned on page top, Zernike aberrations for specific telescope systems, or a mirror, are commonly given in the form of Zernike aberration term, which is, as illustrated on FIG. 31, a product of Zernike orthonormal (normalized to unit variance) polynomial and Zernike expansion coefficient (in effect the RMS wavefront error in units of the wavelength). While formally there is no limit to the number of these terms  or modes  that can be used to describe wavefront structure, relatively smooth wavefronts typically produced by telescope optics are well described by a limited number of Zernike modes. The terms are routinely referred to as "Zernike coefficients" by the amateurs (not seldom, informally, by nonamateurs as well), which is formally incorrect. Zernike (expansion) coefficient is a part of Zernike term; the coefficient equals the RMS error, for optical systems usually given in units of wavelength. An expanded set of Zernike polynomials includes any chosen number of higherorder terms, in addition to the lowerorder terms; in raytracing reports, they are often given in the form of a simple designation zi, with the subscript i indicating term's ordering number, and referred to as Zernike coefficients. This notation is inappropriate, since given values are for the Zernike term, which is always denoted by a capital letter, as opposed to the Zernike (expansion) coefficient  nominally equaling the RMS wavefront error  which is denoted with a small letter. The first term is always piston  an aberration term associated with chief ray, which only constitutes an aberration in systems with two or more pupils differing in phase. It is normally followed by terms expressing lowerorder aberrations, and then those for higherorder forms. Every aberration term except those with full rotational symmetry (defocus and spherical aberration) has two forms, one with cosine, and the other with sine of θ. The sine form effectively rotates wavefront pattern by 90/m degrees counterclockwise (m being, as before, the determinant of angular meridional frequency of the Zernike wavefront mode) with respect to its cosine form, producing wider variety of shapes needed to model asymmetric wavefronts. Specific terms in a set of Zernike polynomials can vary, according to its purpose. Order of terms (or term expansion) is based on the polynomial ordering number. This number is used for a simplified, singleindex term notation. There are several different definitions of the ordering number, with somewhat different forms of term expansion. For evaluating optical systems, one that is commonly used is based on the set of Zernike coefficients defined by J. Wyant (used in OSLO raytracing software). The full set lists 48 Zernike terms; these are the first 15:
TABLE
6: WYANT ZERNIKE
TERMS EXPANSION (first 15) AND RELATION TO STANDARD ABERRATIONS
[1]
Wavefront error in units of the peak aberration coefficient (S,
C,
A and
P for lowerorder spherical
aberration, coma, astigmatism and defocus, respectively); ρ is the
pupil radius normalized
to 1 and θ is the
pupil angle.
Functions w/o
asterisk give PV wavefront error. Functions with the asterisk give
relative wavefront deviations form zero mean circle; they equal peak (i.e. one
half PV) wavefront error, except for balanced primary spherical aberration, where
zero mean splits the PV error in (2/3):(1/3) proportion, as shown below. For instance,
for the peak aberration coefficient S=0.00055mm, the maximum negative deviation given by
the peak wavefront aberration relation
WP=(ρ4ρ2+1/6)S,
is 1/12 (in units of 550nm wavelength, thus with S=1) for ρ=√0.5, and the maximum positive deviation,
for ρ=0
and ρ=1 is 1/6, for λ/4 PV error (at paraxial focus, WPV=Sρ4,
or 1 wave). Note that Zernike term Zi in columns 2 and 4 is often given in inappropriate smallletter notation (which is appropriate for denoting the coefficient), instead of the proper capitalletter notation; twoindex notations at left (column 6, next to the wavefront map), relate Wyant's notation to the ANSI standard indexing scheme, with positive top index m indicating cosine, and negative indicating sine function. Numerical value of the Zernike term Z is given by a product of its polynomial P for ρ=1 and θ=0 for polynomials containing pupil angle θ, normalization factor N and Zernike coefficient z, thus Z=PNz. And since P iz always 1 for ρ=1, Z=Nz. The Zernike coefficient z is nominally equal to the RMS wavefront error, except that it can be negative, as well as positive. Its sign is determined by the orientation of the wavefront point for ρ=1 (edge) relative to zero mean. For instance, primary spherical aberration shown on FIG. 32 could be either under or overcorrection, depending on the direction of light (righttoleft and lefttoright, respectively). As shown, light travels righttoleft, making it undercorrection, and the coefficient is positive. For overcorrection it would have been negative. The sign is only relevant for orientation of the actual wavefront deviation (PV), which can be obtained from the coefficient. The RMS wavefront error, equal to the coefficient in its absolute value (i.e. RMS=z), is always positive.
Following table shows the relation between the value of Zernike term (Z)
and
Zernike coefficient (z) to RMS and PV wavefront error for primary
and secondary aberrations, for the "diffraction limited" RMS wavefront error 0.0745
(hence for z=0.0745) in units of wave. Both Zernike term Z and coefficient
z can
be numerically positive or negative, so they are given as absolute values. Their sign
is always the same. It
does not affect the RMS error, which is always positive, but does determine the
sign for the peak deviation from zero mean, i.e. indicates the
direction of deviation with respect to zero mean.
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 6ρ46ρ2+1 is zero). Thus the PV error is 1.5Nz for spherical aberration, and 2Nz 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 PV 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 PV 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.
