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3.5. Aberration function
▐
3.5.2. Zernike (orthogonal)
aberrations
► 3.5.1. Wavefront aberration function, Seidel aberrationsA brief look at the aberration function helps clarify basic terminology often used with primary aberrations. The aggregate P-V wavefront error W at the Gaussian image point for five primary monochromatic aberrations - also known as Seidel aberrations, after Philipp Ludwig von Seidel, German mathematician whose calculation method first described them in 1857 - for zero defocus, and exit pupil point coordinates (ρ,θ) is given by:
W(ρ,θ) = s(ρd)4
+ cα(ρd)3cosθ
+ aα2(ρd)2cos2θ
+ uα2(ρd)2
+ gα3(ρd)cosθ
(5)
where s, c, a, u
and g are aberration coefficients for spherical aberration,
coma, astigmatism,
field curvature and distortion, respectively,
α
is the field angle,
ρd
is the height in the pupil, with d being the nominal pupil radius
and
ρ
the relative (0 to 1) height in the pupil, and θ
the pupil angle (absent in radially symmetrical aberrations, like
spherical), determining pupil coordinate at which the image point
originates. Since the sum of the powers in
α
and d
terms
is 4, they are also called
4th-order
wavefront aberrations.
For the corresponding transverse ray aberration form, the sum of these
two powers is 3
- for instance, it is (ρd)3/4ƒ2
3α(ρd)2/4ƒ
and ρdα2
for spherical aberration (diameter, paraxial focus, half as much at the best
focus), tangential coma and astigmatism (diameter, best focus),
respectively - so
these are called
3rd-order transverse ray
aberrations (ƒ
is the focal length, while the term in
ρ shows how the aberration
varies with the ray height in the pupil). The next higher
aberration order are 6th-order wavefront and 5th-order ray aberrations
(as mentioned, they are also called secondary, or Schwarzschild
aberrations).
The first three primary aberrations -
spherical, coma and astigmatism - result from
deviations in the wavefront form from spherical. Consequently, their effect
is deterioration in the quality of point-image. The last two - field curvature and distortion
- are image-space aberrations, resulting from deviations in wavefront
radius or orientation (tilt), respectively.
The only primary aberration independent of
the point height in image plane is spherical aberration - it remains
constant across the entire image field. The two aberrations that are
independent of
pupil angle θ
- spherical and field curvature - are symmetrical about the pupil center. In
other words, their property is identical in any given direction from the
point image center (spherical aberration), or from the image field center
(field curvature).
In terms of peak
aberration coefficients, Eq. 5 can be written as:
with the peak aberration coefficient being
either a peak or peak-to-valley (P-V) wavefront error of the aberration,
as explained in more details with each specific aberration. In terms of
Seidel aberration calculation, the
aberration function takes the form:
with S', C', A',
U' and G' being the Seidel sums for spherical
aberration, coma, astigmatism, field curvature and distortion (usually
denoted by SI,
SII,
SIII,
SIII+SIV
and SV),
P being the Petzval sum (denoted by SIV),
and h the point image height in the Gaussian image space
normalized to the maximum object height hmax=1.
Obviously, for the maximum height hmax,
the function becomes
which puts Seidel sums in a direct
relationship with the peak aberration coefficients from Eq. 5.1.
Seidel sums are directly related to the corresponding linear transverse
aberration:
paraxial ray spot radius for spherical aberration is given with S'F2, 1/3 of the comatic
blur (sagittal coma) with C'F and radius of the smallest blur for astigmatism
with A'F, F being the focal ratio (spot
radius resulting from field curvature and image displacement caused by
distortion are also a product of the Seidel sum and the F
number). Likewise, transverse aberrations are directly related to the peak aberration
coefficients from Eq. 5.1, with rs=8SF2,
cts=2CF
and ra=2AF
as the radius of paraxial blur for spherical aberration, sagittal coma,
and radius of the smallest astigmatic circle, respectively.
The aberration function can also be
expressed in terms of Seidel coefficients as:
which puts the Seidel coefficients B,
F, C (not to be confused with the coma peak aberration
coefficient C), D and E, for spherical aberration, coma,
astigmatism, field curvature and distortion, respectively, in a direct
relationship with the primary aberration coefficients given in Eq. 5.
This is valid for the stop at the surface, that is, for the entrance and
exit pupil coinciding. For displaced stop, Seidel coefficients change as
a result of the exit pupil magnification factor m≠1, with the entrance
pupil diameter d replaced by d/m.
Note that Eq. 5-5.4 define primary
aberration at paraxial focus - not the best focus location. For all
three point-image quality (as opposed to image form) aberrations - spherical, coma and astigmatism - best, or
diffraction focus doesn't coincide with the Gaussian image point (i.e.
paraxial focus).
Aberrations evaluated at their best focus location are called
orthogonal, or
balanced, as opposed to
classical aberrations above, which are
evaluated at the Gaussian image point (FIG. 29). The significance of
the classical aberration form is that it provides a common reference
sphere, which makes possible direct calculation of the combined effect
of two or more aberrations with respect to it, as given with Eq. 5.
After that, best reference sphere can be determined for the aberrated
wavefront.
FIGURE 29: All three primary
aberrations affecting point-image quality (FIG.
16)
cause diffraction peak (best focus) to shift away from paraxial
(Gaussian) focus. In the past, these aberrations were evaluated at
paraxial (Gaussian) focus, hence the name "classical aberrations". While
they remain a part of optical textbooks, it is best focus
aberrations, called balanced, or orthogonal, that are of
practical importance. "Orthogonal" relates to a characteristic of the calculations used to
extract them; "balanced" refers to balancing the principal primary
aberration with one or more other aberrations in order to have it
minimized. In effect, the shift from
paraxial to best focus location is determining best-fit reference sphere
for the aberrated wavefront.
Table below shows differences in the size of aberration, and other
specific properties at paraxial and best focus for the three point-image
quality aberrations, in terms of the peak aberration coefficient (S,
C, A for spherical aberration, coma and astigmatism,
respectively), normalized height in the pupil ρ (0<ρ<1) and the
pupil angle θ.
Aberrations
ABERRATION FUNCTION P-V
WAVEFRONT ERROR RMS
WAVEFRONT ERROR Gaussian
focus
Diffraction focus Gaussian
focus Diffraction focus Gaussian
focus Diffraction focus SPHERICAL Sρ4 S(ρ4-ρ2) S S/4 2S/√45 S/√180 COMA Cρ3cosθ C[ρ3-(2ρ/3)]cosθ 2C 2C/3 C/√8 C/√72
ASTIGMATISM Aρ2cos2θ Aρ2(cos2θ-0.5) A A A/4 A/√24
TABLE 3: Properties of the three primary point-image quality
aberrations at paraxial and best focus.
For instance, if the peak aberration coefficient S of spherical
aberration is 1, in units of wavelength, the corresponding P-V wavefront
error at Gaussian (paraxial, when on axis) focus is also 1 (wave), since
the appropriate value of ρ is ±1 (i.e. the wavefront deviation
peaks at the edge). At the best focus location, the error peaks at ρ=√0.5,
hence the corresponding P-V wavefront error is -0.25 waves (the minus
sign indicating that it is on the opposite side of the reference
sphere). For coma, maximum P-V wavefront deviation at either of the two
focus location is for ρ=±1 and θ=0, making the error at
the best focus
three times smaller (note that aberration function for coma expresses
only the peak error, with the corresponding P-V error twice as large).
With astigmatism, the error also peaks for ρ=±1 at both focus locations,
but here the aberration function expresses the P-V error at Gaussian
focus, and peak error (half the P-V error) at the best focus.
Strictly talking, classical aberrations
are primary, or Seidel aberrations, whether third-order
transverse ray or fourth-order on the wavefront; when referring to lower-order aberrations at
the best, or diffraction focus, they should be termed balanced
or orthogonal primary aberrations. However, since practically all aberrations
nowadays are
evaluated at diffraction focus, the term "primary aberrations" is used
for balanced primary aberrations most of the time, for simplicity.
There are two main approaches in calculating the aberrations at the best
focus: one, conventional, is based on the
expansion series describing
conic surface. This means that best focus aberration
it describes is that resulting from wavefront properties given
separately for primary and higher-order aberrations. In other words, it
requires adding up corresponding term-components to find out best focus aberration
for the combined aberration when higher-order components are
significant.
The alternative calculation method, based on
Zernike circle polynomials, overcomes this limitation by allowing for the
inclusion of higher-order aberration components in formulating
direct expressions for the combined best focus aberration.
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