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3.5. Aberration function
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4. INTRINSIC TELESCOPE ABERRATIONS
► 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. Zernike polynomials define deviations from zero mean as a function of the point height ρ in the unit-radius circle and its angular circle coordinate θ, which is the setting of a telescope exit pupil, in which the wavefront form is evaluated. The polynomials are orthogonal (mathematically independent) over a circle of unit radius (i.e. normalized to 1). Due to this attribute, these aberration forms are termed orthogonal, or Zernike aberrations. Orthogonality of Zernike polynomials allows expressing separate contributions of various forms of aberrations - including any chosen extent of 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, and m, the exponent in the image height factor hm. For radially symmetric aberrations, like spherical, 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 an even function of θ, and describes wavefront produced by symmetrical systems (i.e. perfect conic surface). In order to describe actual surfaces, or random aberrations (for instance, wavefront error caused by atmospheric turbulence), Zernike polynomials using uneven function of θ, sin(mθ), need to be included.
Denoting Zernike aberration terms simply as
ZN
(usually written as Z
- 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.
Following table gives overview of Zernike aberration form for 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),
while coma includes its principal aberration term (ρ3)
and balancing tilt term (ρ). 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 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;
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,
Prcos(mθ),
is a product of its radial (Pr)
and angular [cos(mθ)] variable;
COLUMN 5: Zernike
aberration term
Z
The above polynomials describe the aberrations of a perfect conical
surface, hence only use the even function of
θ (cos). Wavefront produced by actual surfaces has random
component to it, and needs to include uneven function of
θ (sin) as well. Hence its aberration term is described
by two polynomial form, one given as Prcos(mθ)
and the other as Prsin(mθ).
An expanded set of Zernike polynomials includes any chosen number of
higher order terms; 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. Every
following term - except defocus and spherical aberration - has two
forms, for even and uneven function of
θ. The corresponding coefficients are put in numerical
order: z0 for piston, z1 and z2 for tilt (for even and uneven function of
θ, respectively), z3 for defocus, z4 and z5 for primary
astigmatism, z6 and z7 for primary coma, z8 for spherical aberration, z9
and z10 for elliptical coma, z11 and z12 secondary astigmatism, z13 an
z14 secondary coma, z15 for secondary spherical aberration, and so on.
Zernike
aberration term ZN
describes orthogonal linear wavefront deviation
from its
zero mean value, that is, deviations from the zero value of the
polynomial as a function of change in ρ, as illustrated on FIG. 19
for lower-order spherical aberration. Zernike spherical
aberration term, either for the phase (ΦS)
or wavefront (ZS)
deviation
for lower-order spherical is zero 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 19: 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 directly
related to it phase error on the other. The former is described by the
wavefront aberration term ZN,
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 best focus, for which the zero mean coincides with the plane containing ρ2=0.5-1/√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
dP),
defines the relative phase deviation over the pupil. The standard phase
deviation value
dP
determines its actual nominal
value. The standard phase deviation is related to the expansion coefficient zs
and the RMS wavefront error ω as dP=2πzs=2πω.
Since the phase error Φ(ρ)
is directly caused by the linear wavefront deviation away from the reference
sphere, after replacing
dP
with zs
or ω, the polynomial expresses the 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)=2πZN=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 P-V wavefront error is
given by a sum of the absolute values of two opposite maximum deviations from
zero mean. In the case of lower-order 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 P-V 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 pupil for this RMS error is dP=2π/√180,
and the resulting phase error Φ(ρ)=π/2,
both in radians
(conversion from the linear to phase error, and vice
versa, is
rather direct, with 1 wave of optical path difference corresponding
to 2π radians phase difference).
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(20ρ6-30ρ4+12ρ2-1)zS.
The zero mean is at the plane containing √0.5
zone (for pupil radius normalized to 1) - as well as two others - on the wavefront deviation plot. The P-V
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 P-V wavefront error as W=2√7ω. Since the P-V wavefront error for
lower-order 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 P-V error for given (identical) RMS wavefront error relates to
that of the balanced 6th order aberration as 1.5√5/2√7.
|
(ρ,θ)),
and the appropriate Zernike expansion coefficients as
z