11.2 The Petzval Theorem

then, that the radius of curvature of the image of a plane object with r1 ¼ 1, is given by

It should be noted that a positive value of r corresponds to a negative sag, or an inward-curving image. Hence the sag of the curved image of a plane object, in the absence of astigmatism, will be given by

This is the famous Petzval theorem, and we shall have many occasions to refer to it since it is only possible to design a flat-field lens free from astigmatism by reducing the Petzval sum; thus the Petzval theorem dominates the entire design processes for flat-field photographic lenses.

The quantity under the summation in these different expressions is called the Petzval sum, and the radius of curvature of the image is evidently the reciprocal of the Petzval sum. Another useful term is the Petzval ratio, which is the ratio of the Petzval radius to the focal length of the lens. It is given by

r0=f 0 ¼ 1=f 0 S

where S is the Petzval sum. Note the reciprocal relationship here. A long focal length lens tends to have a small Petzval sum, while the sum is large in a strong lens of short focal length.

11.2.1Relation Between the Petzval Sum and Astigmatism

It can be shown6 that at very small obliquity angles the tangential astigmatism— that is, the longitudinal distance from the Petzval surface to the tangential focal line—is three times as great as the corresponding sagittal astigmatism. Thus, if the astigmatism in any lens can be made zero, the two focal lines will coalesce on the Petzval surface. In all other cases the locus of the tangential foci at various obliquities is called the tangential field of a lens, and similarly for the sagittal field. As the Petzval surface in most simple lenses is inward-curving, it is often possible to flatten the tangential field by the deliberate introduction of overcorrected astigmatism, leaving the sagittal image to fall between the Petzval surface and the tangential image. However, when designing an “anastigmat” having a flat field free from astigmatism, it is necessary to reduce the Petzval sum drastically.

If it is necessary to design a lens having an inward-curving field to meet some customer requirement, the astigmatism can easily be removed and the Petzval

300 The Oblique Aberrations

sum adjusted to give the desired field curvature. On the other hand, if the field must be backward-curving, it is difficult to avoid an excessive amount of overcorrected astigmatism. It is worth noting that in some types of lens, if the Petzval sum is made too small the separation between the astigmatic fields becomes excessively large at intermediate field angles.

Many decades ago, lens designers taught that the tangential astigmatic field should be flattened to obtain the smallest spot size.7,8,9 This can be easily under-

stood by considering Eqs. (4-6) and (4-7) and assuming that all aberration coefficients are zero other than primary astigmatism ðs3Þ and Petzval ðs4Þ. The sagittal and tangential astigmatic ray errors in the paraxial image plane are

contemplate are a flat sagittal field, a flat tangential field, and equally balanced fields about the paraxial image plane. For a flat sagittal field, s3 þ s4 ¼ 0 or s3 ¼ s4, which means that the residual tangential astigmatism in the paraxial image plane is 3s3 þ s4 ¼ 2s4. When the tangential field is flat, 3s3 þ s4 ¼ 0, which implies that s3 ¼ s4=3 and the residual sagittal astigmatism is 2s4=3. When the errors are balanced, the tangential astigmatism is equal to the negative of the sagittal astigmatism, or s3 ¼ s4=2.

In the balance-fields case, the values of the residual sagittal and tangential astigmatism are observed each to be smaller than the residual values of the prior two cases. This might lead one to select this condition as the optimal minimum spot size10; however, such a conclusion is erroneous.11 It is a general practice by lens designers to adjust the astigmatic surfaces such that the tangential field is flat and then adjust the position of the image plane to the location of the smallest blur at the edge of the field. The definition of the imagery is relatively uniform over the whole image area. In a balanced-field case, the image definition is quite superior in the central region of the image to that of the flat tangential field case, and inferior in the outer portions of the imagery.12 As B. K. Johnson stated, “It therefore depends much on the requirements for which the lens is to be used, as to which criterion is to be adopted.”

11.2.2 Methods for Reducing the Petzval Sum

There are several methods by which the Petzval sum can be reduced, and one or more of these appear in every type of photographic objective. These methods can also be applied to a wide variety of optical systems.

A Thick Meniscus

If we have a single lens in which both radii of curvature are equal and of the same sign, the Petzval sum will be zero, while the lens power is proportional to the thickness. Cemented interfaces in such a lens have very little effect on the

Figure 11.11 Flattening of tangential field using concentric lens where lens was designed to have flat sagittal field. (a) Initial design with flat sagittal field; (b) Tangential field bought into coincidence with the sagittal and Petzval curvatures using the concentric field flattener.

Figure 11.11a. With the introduction of the concentric field flattener lens, the tangential (T) field curvature was brought into coincidence with the Petzval (P) and sagittal (S) field curvatures as depicted in Figure 11.11b. A substantial improvement in resolution uniformity and contrast over the field of view was obtained. An alternative location for the concentric lens is to place it between the front and rear elements. In this case, the radii of the concentric lens are centered at the focus location of the front element.

A concentric field flattener lens may be introduced in other than the image space of a lens; however, this can shift the spatial location of the image plane. Another form of the concentric field flattener can be realized by considering a solid glass plate placed in image space, as illustrated in Figure 11.12a, which has been shown in Sections 3.4.4 and 6.4 to shift the image location and introduce aberrations.16 A concentric air lens17 is now formed by removal of the middle section of the glass plate—that is, a plano-concave lens followed by a convex-plano lens as illustrated in Figure 11.12b. The internal surface curvatures are centered on the image location that would occur should the glass plate have contained the image. The design of a lens being combined with this concentric field flattener element should include the aberrations resulting from a glass plate having a thickness equal to the distance between the plano surfaces of the concentric field flattener lens. The air concentric field flattener lens has a manufacturing advantage over the form shown in Figure 11.9 since it is more difficult to colocate the centers of the surfaces of a concentric lens element.

Another field flattener approach18 uses a concentric shell centered about the stop (convex toward the image). The beam passing through this lens has the chief ray always perpendicular to the surfaces of the shell so its induced

(a)

Glass plate

Figure 11.12 (a) Image shift caused by glass plate. (b) Creation of concentric air lens from the glass plate.

aberrations are constant as a function of field angle. As it does have some negative power, the shell affects the Petzval sum. It also shifts the image somewhat.

PROBLEM: For a concentric field flattener lens, show that spherical aberration, tangential and sagittal coma, axial color, and lateral color are zero, and that distortion and sagittal field curvature are unaffected.

PROBLEM: Consider the field diagrams shown in Figure 11.11 and explain the astigmatic aberrations depicted in both using Seidel aberration coefficients (s3 and s4).

A New-Achromat Combination

By 1886, Abbe and Schott in Jena, Germany, had developed barium crown glasses having just the required property to reduce the Petzval sum, and these glasses were immediately adopted by Schroder in 1888 in his Ross Concentric lens.19 These glasses provided the sought after method for controlling the Petzval sum by using a crown glass of low dispersion and high refractive index in combination with a flint glass of higher dispersion and a low refractive index. This is precisely opposite to the choice of glasses used in telescope doublets and other ordinary achromats. Lenses of this type are therefore known as “new achromats.” They have been used in the Protar (Section 14.4) and many other types of photographic objectives.