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3.5. Aberration function   ▐    3.5.2. Zernike (orthogonal) aberrations
 

3.5.1. Wavefront aberration function, Seidel aberrations

A 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)3/42 3α(ρd)2/4 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 ( 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:

W(ρ,θ) = Sρ4 + Cρ3cosθ + Aρ2cos2θ + Uρ2 + Gρcosθ              (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(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. 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.
    For classical primary spherical aberration, balancing aberration is defocus, numerically equal to the amount of spherical aberration (expressed as the peak aberration coefficient), but of the opposite sign. Resulting aberration - now balanced primary spherical aberration - at the best or diffraction focus (B') is only 1/4 of the primary spherical aberration at Gaussian, or paraxial focus (P'). The gray shaded areas do not represent the wavefront itself (W), rather its grossly exaggerated deviation from a perfect reference sphere (B), centered at the best focus (B'). The deviation is zero at the center and the edge, reaching the maximum at the 0.707 zone. While the actual wavefront doesn't change its shape, the wavefront deviation plot changes when it presents deviation of the actual wavefront from another reference sphere, such as the one centered at paraxial (P') or marginal (M') focus.
     Similarly, balancing aberration for classical primary coma is wavefront tilt, which is practically a rotation of the Gaussian reference sphere (G) centered at the Gaussian image point (G') around its vertex, so that better fitting sphere to the tilted wavefront is found. The new reference sphere (B) is centered at the best focus (B'), and the wavefront error for that point - the focal point for balanced primary coma - is only 1/3 of the error at the Gaussian focus (classical coma error). Unlike spherical aberration, where the point of peak diffraction intensity (best focus) is found in another plane in the image space, best focus for balanced coma is in the same plane (practically, considering very small angle of rotation) as the classical focus.
     Finally, balancing aberration for classical primary astigmatism is defocus. Shown to the left is astigmatism of a concave mirror, for which the wavefront section in the tangential (vertical) plane (T), being of smaller diameter, focuses shorter than the wavefront section in sagittal (perpendicular to it) plane (S), with the intermediate wavefront sections focusing in between. Aberration balancing is accomplished by defocusing to the mid point between sagittal (S') and tangential (T') foci, where is located best focus. This point becomes center of curvature for the perfect reference sphere B for balanced primary astigmatism. While the P-V error remains identical to that at either tangential or sagittal foci, the RMS error at the best focus is smaller by a factor 0.82 (note that cylindrical wavefront deviations at T' and S' are from the respective reference spheres centered at those points).

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

4

S(ρ42)

S

S/4

2S/√45

S/√180

COMA

3cosθ

C[ρ3-(2ρ/3)]cosθ

2C

2C/3

C/√8

C/√72

ASTIGMATISM

2cos2θ

2(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.

 
3.5. Aberration function   ▐    3.5.2. Zernike (orthogonal) aberrations

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