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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 phase-contrast microscope) in 1934, can be applied to describe mathematically 3-D 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 surface; their combined sum can produce a large number of more complex surface shapes, describing specific forms of wavefront deviations (aberrations). 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.
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 reference sphere). 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 reference sphere for balanced primary spherical aberration and defocus, while coinciding with the reference sphere for balanced primary astigmatism. coma and balanced 6th/4th order spherical (B). As a result, the form of polynomial is different from that of the balanced 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, deviation from zero mean for primary spherical aberration - whose polynomial only has the radial component - is given by ρ4-ρ2+1/6; thus, its normalization factor is 6, and the corresponding Zernike circle polynomial is 6ρ4-6ρ2+1. Orthogonality of Zernike polynomials - or, put plainly, the possibility to combine as many different surfaces as needed to approximate the form of wavefront deviation with desired accuracy - 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 scaled to unit variance over the circle radius for all aberration forms, which makes them directly comparable in magnitude. The two subscripts identifying Zernike aberration form are n, the order (exponent) in the pupil height factor ρn (it is the highest power of ρ in the polynomial) and m, the exponent in the image height factor hm. For radially symmetric aberrations, like spherical, the variable in θ is absent; since the aberration changes with ρ4, n=4, and since it is independent of the height h in the image space, m=0. 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). Cos(mθ) is a symmetrical function of θ, and describes wavefront deviations produced by systems with a single axis of symmetry (i.e. conic surface). Such deviations have radial symmetry of |cosθ|=|cos(θ+180)|, i.e. the wavefront has identical type of deformation (although it can be of opposite sign) along two diametrically opposite radii (C, above). In order to describe other type of surfaces, or random aberrations (for instance, wavefront error caused by atmospheric turbulence), terms with asymmetrical function of θ, sin(mθ), need to be included (they are called "asymmetrical" not because they are less symmetrical than the cosine term itself, but because using it with the cosine term creates asymmetrical surfaces). The latter is identical in form to the former, but rotated according to its sine function; their respective values are calculated so that combined with each other produce the specific asymmetrical form of an actual wavefront deviation.
Denoting Zernike aberration terms simply as
ZN
(usually written as
- spherical aberration:
ZS
=
zS√5(6ρ4-6ρ2+1)
= S(6ρ4-6ρ2+1)/6,
with n=4, m=0
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 Zernike aberration term, ωS=zS=ZS/√5, ωC=zC=ZC/√8
and ωA=zA=ZA/√6
(coma and astigmatism for θ=0).
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 the factor
φ
of
standard deviation, i.e.
phase error averaged over the pupil (requirement fulfilled for
low-level aberrations affecting most or all of wavefront area, roughly
below
λ/2 P-V
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(1-o2)2
=
zS(1-o2)2
= ZS(1-o2)2/√5
ωCo= zCo
= ZCo/√8
= ZC(1-o2)(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, for clear circular aperture (aberrations in
aperture with central obstruction are described with Zernike annular
polynomials). The three point-image 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
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 only by combining
with defocus) includes its
principal aberration term ρ6
and two balancing terms, for lower-order spherical and defocus (ρ4
and ρ2,
respectively). The polynomial forms are as given by Mahajan (Optical
Imaging and Aberrations). 1 2 3 4 5 6 ABERRATION n m ZERNIKE ZERNIKE
ABERRATION TERM
RMS WAVEFRONT ERROR Tilt (Distortion) 1 1 ρcosθ 2ρcosθ 2 Defocus 2 0 2ρ2-1 √3(2ρ2-1) 1/√3 Primary spherical 4 0 6ρ4-6ρ2+1 √5(6ρ4-6ρ2+1) 1/√5 Secondary spherical
(balanced 6th/4th) 6 0 20ρ6-30ρ4+12ρ2-1 √7(20ρ6-30ρ4+12ρ2-1) 1/√7 Primary coma 3 1 (3ρ3-2ρ)cosθ √8(3ρ3-2ρ)cosθ 1/√8 Secondary coma 5 1 (10ρ5-12ρ3+3ρ)cosθ √8(10ρ5-12ρ3+3ρ)cosθ 1/√8 Primary astigmatism 2 2 ρ2cos2θ √6ρ2cos2θ 1/√6 Secondary astigmatism 4 2 (4ρ4-3ρ2)cos2θ √10(4ρ4-3ρ2)cos2θ 1/√10
TABLE 3: Zernike circle polynomials for
balanced (best focus) aberrations listed in column 1; "primary" refers
to 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, third and fifth-order, respectively);
COLUMN 2: n is the
aberration function exponent in the (normalized to 1) pupil height factor ρ;
COLUMN 3: m is the
aberration function exponent in the image height factor h=αƒ,
α
being the field angle and
ƒ
the system
focal length; COLUMN 4:
General 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 5: Zernike
aberration term for circular aperture, which equals Zernike expansion
coefficient
Z
Each separate polynomial in the above table describes the aberrations of a perfect conical
surface, hence only use even function (cos) of
θ (wavefront produced by actual surfaces may have
significant asymmetric
component, and need to include uneven function in sinθ as well).
Zernike aberrations for specific telescope systems are commonly given in
the form of Zernike aberration coefficients, a product of Zernike
aberration term and the RMS wavefront error in units of the wavelength
(i.e.
|
Zernike polynomials define
deviations
from
zero mean as a function of the radial point height ρ
in the unit-radius circle and its angular circle coordinate θ, which is the
setting of a telescope 
and referred to as "Zernike coefficient"),
and the appropriate Zernike expansion coefficients as
z
(ρ,θ)/ω=[2(n+1)/(1+δ