telescopeѲptics.net
▪
▪
▪
▪
▪▪▪▪
▪
▪
▪
▪
▪
▪
▪
▪
▪ CONTENTS
◄
3.5.2. Zernike aberrations
▐
3.5.3. Zernike expansion schemes
► 3.5.2. (cont.) Zernike aberration terms
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 single-index notation, as Zj, informally - and incorrectly - referred to as "Zernike coefficients"), and 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 point-image 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(6ρ4-6ρ2+1)
= |S|(6ρ4-6ρ2+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/P-V wavefront error, or Zernike expansion coefficients,
which can be numerically negative.
Similarly, conversion for defocus is
ZD= zD√3(2ρ2-1)= 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
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 in telescopes, 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
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 lower-order spherical and defocus (ρ4
and ρ2,
respectively). The polynomial forms are as given by Mahajan (Optical
Imaging and Aberrations).
ZERNIKE CIRCLE POLYNOMIALS FOR COMMON ABERRATIONS (BEST
FOCUS LOCATION) I II III IV V VI ABERRATION n m ORTHOGONAL ORTHONORMAL
ZERNIKE
RMS ERROR Tilt
(Distortion) 1 1 ρcosθ 2ρcosθ [2ρcosθ]z11 Ztd/2 Defocus 2 0 2ρ2-1 √3(2ρ2-1) [√3(2ρ2-1)]z20 Zd /√3 Spherical primary 4 0 6ρ4-6ρ2+1 √5(6ρ4-6ρ2+1) [√5(6ρ4-6ρ2+1)]z40 Zs1/√5 secondary (balanced
6th/4th) 6 0 20ρ6-30ρ4+12ρ2-1 √7(20ρ6-30ρ4+12ρ2-1) [√7(20ρ6-30ρ4+12ρ2-1)]z60 Zs2 /√7 Coma primary 3 1 (3ρ3-2ρ)cosθ √8(3ρ3-2ρ)cosθ [√8(3ρ3-2ρ)cosθ]z31 Zc1/√8 secondary 5 1 (10ρ5-12ρ3+3ρ)cosθ √12(10ρ5-12ρ3+3ρ)cosθ [√12(10ρ5-12ρ3+3ρ)cosθ]z51 Zc2/√12 Astigmatism primary 2 2 ρ2cos2θ √6ρ2cos2θ [√6ρ2cos2θ]z22 Za1/√6 secondary 4 2 (4ρ4-3ρ2)cos2θ √10(4ρ4-3ρ2)cos2θ [√10(4ρ4-3ρ2)cos2θ]z42 Za2/√10
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
non-symmetrical 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 non-amateurs 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
higher-order terms, in addition to the lower-order 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 lower-order aberrations, and then
those for higher-order 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,
single-index 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 2
3 4 5 6
#
ZERNIKE
Zernike RMS
Aberration WF Map[2]
Name
Standard aberration function[1]
Paraxial focus
Best focus
0
Z0
1
Z0
piston - -
1
Z1
ρcosθ
Z1
/2
distortion/tilt
ρcosθ
ρcosθ
2
Z2
ρsinθ
Z2
/2 - -
3
Z3
2ρ2-1
Z3
/√3
defocus/field curvature
ρ2 ρ2
4
Z4
ρ2cos2θ
Z4
/√6
primary astigmatism ρ2cos2θ ρ2(cos2θ-0.5)
*
5
Z5
ρ2sin2θ
Z5
/√6 - -
6
Z6
(3ρ3-2ρ)cosθ
Z6
/√8
primary coma
ρ3cosθ
* (ρ3-2ρ/3)cosθ
*
7
Z7 (3ρ3-2ρ)sinθ
Z7
/√8 - -
8
Z8
6ρ4-6ρ2+1
Z8
/√5
BALANCED ρ4 ρ4-ρ2
9
Z9
ρ3cos3θ
Z9
/√8
elliptical coma
ρ3cos3θ
* ρ3cos3θ
*
10
Z10 ρ3sin3θ
Z10
/√8 - -
11
Z11
(4ρ4-3ρ2)cos2θ
Z11
/√10
secondary
ρ4cos2θ (ρ4-0.75ρ2)cos2θ
*
12
Z12 (4ρ4-3ρ2)sin2θ
Z12
/√10 - -
13
Z13
(10ρ5-12ρ3+3ρ)cosθ
Z13
/√12
secondary coma ρ5cosθ (ρ5-1.2ρ3+0.3ρ)cosθ
*
14
Z14 (10ρ5-12ρ3+3ρ)sinθ
Z14
/√12 - -
15
Z15
20ρ6-30ρ4+12ρ2-1
Z15
/√7
secondary
spherical aberration ρ6
ρ6-1.5ρ4+0.6ρ2
[1]
Wavefront error in units of the peak aberration coefficient (S,
C,
A and
P for lower-order 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 P-V wavefront error. Functions with the asterisk give
relative wavefront deviations form zero mean circle; they equal peak (i.e. one
half P-V) wavefront error, except for balanced primary spherical aberration, where
zero mean splits the P-V 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=(ρ4-ρ2+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 P-V error (at paraxial focus, WP-V=Sρ4,
or 1 wave).
Note that Zernike term Zi
in columns 2 and 4 is often given in inappropriate
small-letter notation (which is appropriate for denoting the coefficient), instead of the proper
capital-letter notation; two-index 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 (right-to-left and
left-to-right, respectively). As shown, light travels right-to-left, 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 (P-V), 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 P-V 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 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.
|