In Figure 4.5, the Gaussian image height is shown in a dashed box as the linear portion of the aperture independent comatic aberration. This should make sense in that comatic aberrations are considered associated with variation in magnification with respect to object height and entrance pupil. It also fills out the H expansion sequence in the aperture independent comatic aberrations. Hence, distortion of the image can be considered as the aberration of the principal ray and is defined by the following equation.

DISTðHÞ ¼ Yð0; 00; H; xÞ GIHðH; xÞ

(4-10)

¼ s5H3 þ m12H5 þ t20H7

where the Gaussian image height is GIHðH; xÞ ¼ GIHðHÞ þ DFyðr; 00; xÞ. Distortion is considered negative when the actual image is closer to the axis

than the ideal image, and positive distortion is the converse. This physically means that the image of a square suffering negative distortion will take on a barrel-like appearance and is referred to as barrel distortion. In the case of positive distortion, the image takes on a pincushion-like appearance and is referred to as pincushion distortion. To reiterate, distortion is an aperture independent comatic aberration. For most lenses, distortion beyond the third-order term is minimal.

4.3.6 Selection of Rays for Aberration Computation

Table 4.1 presents the five rays necessary to compute the astigmatic and comatic aberrations for a particular set of ðr; HÞ. It should be noticed that the first three rays all contained in the meridional plane. The remaining two rays are skew rays. The sagittal astigmatism is the only aberration to utilize x-coordinate ray data.

Table 4.1

Table of Rays Required to Compute the Astigmatic and Comatic Aberrations

4.3.7 Zonal Aberrations

Due to the great labor and tedious nature of tracing rays, early lens designers carefully chose the rays to trace, which were primarily meridional rays and lesser sagittal rays. Even if the meridional rays and sagittal rays came to perfect focus, these designers understood the importance of tracing a few general skew rays and the significant increase in computational labor required to do so. Why was this important? At the time, the designers had little in-depth theoretical knowledge to reach this conclusion; however, their experience was that such a ray trace was necessary to assure the quality of the design.

Looking at the 37 optical aberration coefficients through the seventh order, it can be shown that not all of the aberration coefficients are accounted for by the determination of spherical aberration, coma, astigmatism, and distortion. The “missing” aberration coefficients are m9; t9; t14; and t17: To account for these coefficients, several additional defect definitions are added to those already discussed.14 These are denoted as tangential and sagittal zonal astigmatism, and tangential and sagittal zonal coma, which use evaluation-plane ray intercept data from the two rays having coordinates of ðr; 45o; H; xÞ and ðr; 135o; H; xÞ in addition to intercept data from the marginal ray.

4.3.8 Tangential and Sagittal Zonal Astigmatism

The defining equations for tangential zonal astigmatism and sagittal zonal

As explained previously, astigmatic aberrations computed using these equations inherently have the defocus contribution subtracted thereby yielding

the preceding aberration definitions. Just as in any design process, the designer needs to appropriately select object heights and entrance pupil coordinates for the design task at hand. In addition, the astigmatic aberrations were formulated to remove dependence on defocusing of the image plane, with respect to the Gaussian image plane. The comatic aberrations are inherently independent of image plane location. It is also helpful to have an estimation of the higher-order contributions to the aberration coefficients. Conrady was perhaps the first to derive equations expressing the higher-order astigmatic and comatic aberrations.

Referring to Figure 4.5, the field-dependent astigmatic aberrations are divided into linear and nonlinear terms with respect to entrance pupil radius. The tangential astigmatism previously defined contains both linear and nonlinear terms. The nonlinear term is typically referred to as oblique spherical aberration and is a particularly onerous aberration. It is actually rather simple to compute oblique spherical aberration by subtracting the linear term of tangential astigmatism from the total tangential astigmatism term. The linear term is determined by computing the tangential astigmatism for an entrance pupil radius r0 much smaller than the radius r being used to calculate the tangential astigmatism itself. The linear term is appropriately scaled and subtracted from the tangential astigmatism to obtain the tangential oblique spherical aberration. This is expressed by the following equation.

where r0 r:

In a like manner, the sagittal oblique spherical aberration is computed using the following equation.

where r0 r:

Obviously the linear terms

for the tangential and sagittal astigmatism, respectively, can be utilized in the design process.

Similarly, the comatic aberration has an aperture dependent set of aberrations, namely, linear coma and nonlinear coma. The nonlinear tangential coma is found by subtracting the appropriately scaled linear tangential coma from the