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3.5. Aberration function
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3.5.2. Zernike aberration coefficients
► 3.5.2. Zernike aberrations An alternative way of describing best focus telescope aberrations are Zernike circle polynomials. These polynomials, introduced by the Dutch scientist Fritz Zernike (Nobel prize laureate for the invention of phasecontrast microscope) in 1934, can be applied to describe mathematically 3D wavefront deviation from what can be constructed as a plane  i.e. unit circle  of its zero mean, defined as a surface for which the sum of deviations on either side  opposite in sign one to another  equals zero. Each polynomial describes specific form of surface deviation; their combined sum can produce a large number of more complex surface shapes, that can be fit to specific forms of wavefront deviations (aberrations). In principle, by including sufficient number of Zernike polynomials (commonly referred to as terms), any wavefront deformation can be described to a desired degree of accuracy. The usual way of applying Zernike terms is to the specific wavefront shape, which is "decomposed" to a needed number of terms in order to determine: (1) the main forms of contributing deviations, and (2) the overall magnitude of deformation. For simple aberration forms, such as pure Siedel aberrations, a single polynomial suffices. Describing more complex aberrations, such as, for instance, seeing error, as well as wavefronts formed by actual (i.e. imperfect) surfaces, requires an expanded set of Zernike polynomials.
Zernike polynomials define
deviations
from
zero mean as a function of the radial point height ρ
in the unitradius circle and its angular circle coordinate θ, which is the
setting of a telescope exit pupil, in which the wavefront form is
evaluated (FIG. 30, 1). In
polar and Cartesian coordinates, respectively, the radial component is ρ2=x2+y2,
with 0≤ρ,x,y≤1.
The common convention for the angular coordinate θ varies
with the field; in ophthalmology, it is
counterclockwise from x+ toward y+ axis (OSA recommended),
thus ρ=x/cosθ=y/sinθ. In general optics, it is often
different. Malacara's convention is clockwise from
y+ to x+ axis, thus ρ=x/sinθ=y/cosθ, and Mahajan's
convention (Optical imaging and aberrations) applied here to conventional aberration functions is
counterclockwise from y+ to x, hence with the same
radialtoangular relations as Malacara's. The polynomials are orthogonal (i.e. their values change independently,
as illustrated on FIG. 30, 1) over the circle
of unit radius. Due to this
attribute, these aberration forms are termed orthogonal,
or Zernike aberrations.
As mentioned, zero mean is defined as a surface for which the sum of wavefront deviations to either side is zero. That is important conceptual difference vs. standard wavefront error, which expresses deviations from a reference sphere (also commonly constructed as a circle). Hence the polynomial, which is a product of its radial variable in ρ and angular variable in θ, has zero value at the intersection of the wavefront and its zero mean. Zero mean differs from the reference sphere for balanced primary spherical aberration and defocus, while coinciding with it for balanced primary astigmatism. coma and balanced 6th/4th order spherical (FIG. A, 2). As a result, the form of polynomial is different from the classical aberration function for the former three, while identical (except for the normalization factor) for the latter two. The polynomial normalization factor fulfils the formal requirement that the radial polynomial portion equals 1 for ρ=1. For instance, the deviation from zero mean for primary spherical aberration  whose polynomial only has the radial component  is given by ρ4ρ2+1/6; thus, its normalization factor is 6, and the corresponding Zernike circle polynomial is 6ρ46ρ2+1 (this normalization to unit radius shouldn't be confused with normalization to unit variance, described ahead). Orthogonality of Zernike polynomials creates the possibility to combine as many different surfaces as needed to approximate the form of wavefront deviation with desired accuracy. It allows expressing separate contributions of various forms of aberrations  including any chosen extent of the higher order forms  and obtaining the combined variance as the sum of individual aberration variances. Also, the polynomials can be  and routinely are  scaled to unit variance over the circle radius for all aberration forms, so that their combined form can be determined directly by adding up their expansion coefficients, which determine the specific magnitude for each aberration form. Wavefront is described as a sum of Zernike aberration terms (FIG. 31).
In the nutshell, the normalization factor N is chosen so that a product of the sum of two extreme values of the polynomial (absolute values, determining the relative magnitude of PV deviation) and normalization factor equals the PVtoRMS wavefront error ratio for the aberration. Hence, multiplying this product with the expansion coefficient  which equals the RMS error for given aberration  yields the PV wavefront error corresponding to the coefficient. For any value of the polynomial for given pupil coordinate ρ, a product with the normalization factor and expansion coefficient yields, as already mentioned, the wavefront deviation from zero mean for that particular pupil coordinate.
EXAMPLE: Plots for orthogonal and orthonormal
Zernike polynomials vs. those of the standard aberration function for
primary spherical aberration and coma. All plots for either aberration
represent the same type of function  i.e. form of deviation  the only
difference being in their nominal maxima or position vs. abscissa
(horizontal scale), which represents the pupil, with the pupil radius
ρ normalized to unit ranging from 1
to 1. Function
ƒ(ρ)
 which is the wavefront deviation over pupil (with
θ=0 for coma) 
shows how the aberration changes over the pupil. In general, plots for
Zernike terms have significantly greater amplitude than the
corresponding standard functions, due to the coefficients (integer
multiplier assigned to the variable) being larger. As with the standard aberrations, the wavefront error, either PV (as a direct optical path difference) or RMS, is directly related  although not necessarily identical  to the phase error. The absolute value of Zernike expansion coefficient znm is identical to the RMS wavefront error; since the coefficient does express positive and negative deviations, the sum of coefficients for all Zernike terms used to fit particular wavefront gives its overall RMS wavefront error (i.e. standard deviation), and its square equals the wavefront variance. The two integers identifying Zernike aberration form are n, the highest order (exponent) in the polynomial's radial variable V (analog to the pupil height factor ρn in the standard aberration functions) and m, the angular frequency of meridional variance (nominally identical to the exponent in the image height factor hm in the standard aberration functions). For radially symmetric aberrations, like spherical, the angular variable cos(mθ) or sin(mθ) is absent, thus m=0 (alternately, since it is independent of the height h in the image space, m=0); and, since the aberration changes with ρ4, n=4. For primary coma, which changes with ρ3 and h, n=3 and m=1; since it varies with the point pupil angle θ, it also includes the angular coordinate factor, in the form cos(mθ). Consequently, Zernike aberration terms for primary spherical aberration and coma are denoted as Z and Z, and Zernike expansion coefficients as z40 and z31, respectively (note that according to the above convention, m=1 indicates cosine function, i.e. coma peaks are positioned at a horizontal line; for the vertical orientation, m=1). Likewise, Zernike aberration term and expansion coefficient for primary astigmatism, which changes with the 2nd power in both, pupil and image space (the latter is not formally the basis for indexing, but is numerically correct and convenient), thus with n=2 and m=2 or 2, are Z, or Z, and z22 (which, as any Zernike expansion coefficient, can be numerically positive, or negative, depending on the orientation of deformation). For defocus, which is radially symmetrical like spherical aberration, but changes with the square of pupil height, n=2 and m=0, hence its Zernike term is Z and its expansion coefficient is z20. As mentioned, every Zernike aberration term (or mode) describes specific orthogonal wavefront deviation from its zero mean, that is, deviations from zero value of the polynomial as a function of change in radial coordinate ρ and angular coordinate θ. How Zernike aberration term  i.e. orthonormal polynomial  specifically describes an aberrated wavefront is illustrated on primary spherical aberration (FIG. 32). For simplicity, the polynomial Z is denoted by ZS and the expansion coefficient z40 as zS; the relative linear wavefront deviation from zero mean as W(ρ), with the corresponding phase deviation Φ(ρ); as usual, the RMS wavefront error is ω, with the corresponding phase RMS error analog φ, and standard phase deviation φ=2πφ (the error variance is, by definition, the standard deviation squared). Zernike aberration term, either for the phase (ΦS) or wavefront (ZS, identical to W(ρ), the latter being used to relate the nature of it more directly) deviation for lowerorder spherical is zero when the sum in brackets is zero. This occurs for ρ2=0.5±1/√12, regardless of the size of aberration, since the sum of deviations between these two zonal heights is identical to the sum of deviations over the rest of the wavefront (which are of opposite sign relative to the plane of zero mean).
FIGURE 32: Zernike circle polynomials can be
used to express the two main aspects of wavefront aberrations: linear
deviations away from the reference sphere on one side, and closely
related to it phase error on the other. The former is described by the
wavefront aberration term Z
(here written simply as ZS
for spherical aberration), and the latter by the phase error term Φ(ρ).
The latter expresses orthogonal phase deviation, in
radians, from zero mean plane (Φ(ρ)=0),
over a circle of unit radius ρ. Unlike the standard
wavefront error, which is measured with respect to a reference sphere,
Zernike polynomials express the deviation from
zero mean. Shown to the left is the primary,
4th order spherical
aberration at the best focus, for which zero mean coincides with the plane containing ρ2=0.51/√12
and ρ2=0.5+1/√12
zones. The two phase deviation sums  one to the left,
the other to the right of the zero mean plane  are equal and of opposite signs
(the polynomial itself is zero for these ρ values).
The base polynomial  without the standard
(phase) deviation value φ 
defines relative phase deviation over the pupil. The standard
deviation value φ
determines its actual nominal
value. It is related to the expansion coefficient zs
and the RMS wavefront error ω as
φ=2πφ=2πzs=2πω
(valid for PV error <0.5λ), with φ
being the phase analog to RMS wavefront error (note that unlike the RMS
error ω, Zernike coefficient zs
can be numerically negative; if the wavefront shown converges to the
left  in which case it represents so called "undercorrection"
 the deviation adds to the optical path length of reference sphere,
with the coefficient value being positive, and vice versa).
Since the phase error Φ(ρ)
is directly caused by linear wavefront deviations away from the reference
sphere, after replacing φ
with zs
or ω, the polynomial expresses linear wavefront deviation
from its zero mean, which coincides with the zero mean of the
corresponding phase error. Hence, the two aberration terms relate as Φ(ρ)=2πZs=2πW(ρ),
with the term W being used to denote linear
wavefront error in this site (note that here it is relative to zero
mean, not the reference sphere). It implies that the PV wavefront error is
given by a sum of the absolute values of two opposite maximum deviations from
zero mean. In the case of lowerorder spherical aberration, as can be
seen from the plot, these two
maximum deviations are for ρ=0 or ρ=1, and for ρ=√0.5. So, for the RMS
wavefront error ω=1/√180,
in units of the wavelength, the corresponding PV wavefront error,
given by the sum of either W(0) or
W(1)
and W(√0.5)
is,
as expected, WPV=1.5√5ω=0.25, also in units of the wavelength. The corresponding
standard phase deviation over the pupil for this RMS error is φ=2π/√180,
and the resulting phase error Here, linear wavefront deviation W(ρ), specified by, and equal to the Zernike aberration term, is different form the peak, or PV value given by the standard aberration form, because zero mean does not coincide with the reference sphere. However, for aberrations where the two coincide  such as primary coma and astigmatism  Zernike aberration term equals the wavefront peak, or PV error corresponding to the absolute value of Zernike expansion coefficient, i.e. the wavefront RMS error (Zernike coefficient, unlike RMS wavefront error, can be negative, since its sign identifies the spatial orientation of deformation; the sign is determined by the direction of wavefront deviation from reference sphere, along the axis of aberration: if it adds to the OPL, coefficient is positive, and vice versa  on the above illustration, for wavefront converging to the left, the deviation adds to the OPL, and the sign of coefficient is positive). For instance, Zernike term for primary coma, Z=z31√8(3ρ32ρ)cosθ, has the maximum value of √8z31 for ρ=1 and θ=0, cosθ=1 (i.e. along the axis of aberration). For the diffraction limited RMS value of the expansion coefficient, z31=1/√180 in units of wavelength, it gives Z=1/√22.5, equaling the peak wavefront error, also in units of wavelength (if the coefficient is quoted in linear units, for instance microns, the term expresses the peak wavefront error in microns). The PV error is doubled, since the other, opposite in sign extreme value of the polynomial is identical in its relative magnitude, for θ=180°, cosθ=1. Likewise, Zernike term for primary astigmatism Z=z22√6ρ2cos2θ, with the maximum value of √6z22, also equals the peak wavefront error for any given expansion coefficient (i.e. RMS). Even for defocus, where zero mean and reference sphere do not coincide (FIG. 30, 2), Zernike aberration term Z=z20√3(2ρ21) will equal the peak wavefront error (for ρ=1), because the zero mean splits the maximum wavefront deviation in two halves. As another example, Zernike aberration term for 6th order spherical aberration  the form that is optimally balanced with 4th order spherical  is given by the polynomial ZS = √7(20ρ630ρ4+12ρ21)zS. The zero mean is at the plane containing √0.5 zone (for pupil radius normalized to 1)  as well as two others for which the polynomial is zero  on the wavefront deviation plot. The PV wavefront error is determined by a sum of the absolute values of maximum deviations from the zero mean, which occur for ρ=0 and ρ=1. With ZS=W(ρ), and zS=ω (the RMS wavefront error), this gives the PV wavefront error as W=2√7ω. Since the PV wavefront error for lowerorder spherical aberration, as already mentioned above, is a sum of the deviations for ρ=1 or ρ=0, and ρ=√0.5, it is given by W=1.5√5ω, and its PV error for given (identical) RMS wavefront error relates to that of the balanced 6th order aberration as 1.5√5/2√7. Another interesting property of Zernike aberration terms implicated by FIG. 32 is that the PV/RMS ratio can be expressed as (1+d)N, where d is the maximum relative wavefront deviation from zero mean (as an absolute value) to the side opposite to the reference sphere  which is always in the plane containing the vertex  in units of the deviation from zero mean toward reference sphere, and N is the term's normalization (square root) factor. For most aberrations (all primary aberrations except spherical, as well as all secondary aberrations, including trefoil and spherical), d=1 and the PV/RMS ratio is given by 2N. So for coma, with the normalization factor equaling √8, the PV/RMS ratio is 2√8, and for astigmatism, with normalization factor √6, the PV/RMS ratio is 2√6. For spherical aberration, as shown on FIG. 32, d=0.5 and the PV/RMS=1.5N=1.5√5. As already mentioned, most common conic aberrations can be described with a single Zernike aberration term, with either cosine or sine angular function (the choice only affect wavefront orientation). However, in order to describe wavefronts generated by irregular surfaces  with this qualification applying to some degree to all actual optical surfaces  or random aberrations (for instance, wavefront error caused by atmospheric turbulence), multiple Zernike terms, with both sine and cosine orientations, need to be included. Following page presents in more detail the properties of Zernike aberrations for common lowerorder aberrations, as well as expanded list of Zernike terms  often inappropriately referred to as "Zernike coefficients"  that includes higherorder aberrations as well.
