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3.5. Aberration functionA 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,q) is given by: W = s(ρd)4 + cα(ρd)3cosq + aα2(ρd)2cos2q + uα2(ρd)2 + gα3(ρd)cosq (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 q the pupil angle, 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, 8sƒ(ρd)3, 3α(ρd)2/4ƒ and ρdα2 for spherical aberration (diameter, paraxial focus, half as much at best focus), tangential coma and astigmatism (diameter, best focus), respectively - so these are called 3rd-order ray (or geometric) 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 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 the image plane is spherical aberration - it remains constant across the entire image field. The two aberrations that are independent of the pupil angle q - 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 = Sρ4 + Cρ3cosq + Aρ2cos2q + Uρ2 + Gρcosq (5.1) 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: W = (S'/8)ρ4 + (C'/2)hρ3cosq + (A'/2)h2ρ2cos2q + (U'/4)(A'+P)h2ρ2 + (G'/2)h3ρcosq (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 = (S'/8)ρ4 + (C'/2)ρ3cosq + (A'/2)ρ2cos2q + (U'/4)(A'+P)ρ2 + (G'/2)ρcosq (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 number (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, cs=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 = -(B/4)(ρd)4 + Fα(ρd)3cosq - Cα2(ρd)2cos2q - (D/2)α2(ρd)2 + Eα3(ρd)cosq (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. 5 defines primary aberration for 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 aberration, which are evaluated at the Gaussian image point (FIG. 15). 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 15: All three primary
aberrations affecting point-image quality (FIG. 12)
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 the best focus
aberrations, called balanced, or orthogonal, that are of practical
importance. The name
"orthogonal" comes from the characteristic of calculations used to
extract them; "balanced" refers to balancing a principal primary
aberration with another aberration in order to minimized it - which is, in fact, shifting from the
paraxial to best focus location. Strictly talking, classical aberrations are primary, or Seidel aberrations, whether third-order ray or fourth-order wavefront; when referring to lower-order aberrations at 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. An alternative way of describing orthogonal (best focus) aberrations are Zernike polynomials. Advantage of this approach is that it can be extended to higher order aberrations, as well as random aberrations forms. Denoting Zernike aberration terms simply as ZN (usually written as Φmn(ρ,θ)), and the appropriate Zernike expansion coefficients as zN (usually written as cnm), where the subscript N identifies the corresponding aberration, Zernike form for the 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 ZS≡Φ04
and zS≡c40 This implies that Zernike expansion coefficients zS, zC and zA, equal the corresponding RMS wavefront error ω in terms of peak aberration coefficient, given by ωS=S/√180, ωC=C/√72 and ωA=A/√24 for spherical aberration, coma and astigmatism, respectively.
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