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3.5. Aberration function
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3.5.2. Zernike (orthogonal)
aberrations
► 3.5.1. Seidel aberrations, wavefront aberration functionA brief look at the aberration function helps clarify basic terminology often used with primary aberrations. The aggregate PV 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 4thorder 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 3rdorder 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 6thorder wavefront and 5thorder 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 pointimage. The last two  field curvature and distortion  are imagespace 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: W(ρ,θ) = Sρ4 + Cρ3cosθ + Aρ2cos2θ + Uρ2 + Gρcosθ (5.1) with the peak aberration coefficient being either a peak or peaktovalley (PV) 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: W(ρ,θ) = (S'/8)ρ4 + (C'/2)hρ3cosθ + (A'/2)h2ρ2cos2θ + (U'/4)(A'+P)h2ρ2 + (G'/2)h3ρcosθ (5.2) 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 W(ρ,θ) = (S'/8)ρ4 + (C'/2)ρ3cosθ + (A'/2)ρ2cos2θ + (U'/4)(A'+P)ρ2 + (G'/2)ρcosθ (5.3), 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: W(ρ,θ) = (B/4)(ρd)4 + Fα(ρd)3cosθ  Cα2(ρd)2cos2θ  (D/2)α2(ρd)2 + Eα3(ρd)cosθ (5.4) 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. 55.4 define primary aberration at paraxial focus  not the best focus location. For all three pointimage 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 pointimage 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 bestfit 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 pointimage
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.
TABLE 3: Properties of the three primary pointimage 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 PV 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 PV wavefront error is 0.25 waves (the minus sign indicating that it is on the opposite side of the reference sphere). For coma, maximum PV 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 PV error twice as large). With astigmatism, the error also peaks for ρ=±1 at both focus locations, but here the aberration function expresses the PV error at Gaussian focus, and peak error (half the PV error) at the best focus. Strictly talking, classical aberrations are primary, or Seidel aberrations, whether thirdorder transverse ray or fourthorder on the wavefront; when referring to lowerorder 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 higherorder aberrations. In other words, it requires adding up corresponding termcomponents to find out best focus aberration for the combined aberration when higherorder components are significant. The alternative calculation method, based on Zernike circle polynomials, overcomes this limitation by allowing for the inclusion of higherorder aberration components in formulating direct expressions for the combined best focus aberration.
