tangential coma. The linear tangential coma is determined by computing the tangential coma for a comparatively small object height, that is, H0 H. The equation for the nonlinear tangential coma is given by

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

4.4CALCULATION OF SEIDEL ABERRATION COEFFICIENTS

In 1856, Philip Ludwig von Seidel published his work on a systematic method for computing third-order aberrations and provided explicit formulas. These aberrations are commonly referred to as the Seidel aberrations and are denoted in order of spherical, coma, astigmatism, Petzval (field curvature), and distortion by a variety of symbols in different books and papers such as

(a) s1 though s5; (b) SC, CC, AC, PC, and DC; (c) SI ; SII ; . . . ; and SV ; (d) B, F, C, P, and E; (e) 0a40; 1a31; 2a22; 2a20; and 3a11; and others. When using any computation scheme to determine the Seidel aberrations, care should be taken to understand if the values are coefficients only, transverse aberrations, longitudinal aberrations, or wave aberrations. In the following, a method will be presented for computing s1 though s5 aberration coefficients from simply marginal and principal paraxial ray data. By multiplying these coefficients by the appropriate factor, transverse, longitudinal, and wave aberrations can be obtained although it is often common that the symbols s1 though s5 be used after the transformation to transverse, longitudinal, or wave aberrations.

There are a variety of approaches to derive equations to compute the Seidel aberration coefficients. The method followed here is after Buchdahl, but only the general approach is presented as the details can be easily worked out. By tracing a marginal paraxial ray and a principal paraxial ray, using Eq. (3-2),

at a surface, it can be shown that

ynu ynu ¼ yn0u0 yn0u0

where y and u represent the principal-ray values. This implies that ynu ynu is

a constant across any surface. Using Eq. (3-3), it can be shown that ðynu ynuÞi at the ith surface is equal to ðynu ynuÞiþ1 at the (iþ1)th surface, which means

that the term is also constant within the space between the surfaces. This term is called the optical invariant.

PROBLEM: Using Eqs. (3-2) and (3-3), show that ynu ynu is invariant across surfaces and in the space between surfaces.

Consider now an object located in the meridional plane having height Hy ¼ h and Hx ¼ 0 since the object is aberration free. For a stigmatic optical system, the paraxial and real ray image heights must be identical and related to the object height by the magnification, that is, h0 ¼ mh ¼ mHy. As stated previously, an imperfect system will suffer some ray aberration and the transverse ray aberration is given by ey Hy0 h0 and ex Hx0 . Now trace two rays from the object with one starting at the base of the object and the other at the object’s head. Using a subscript o to designate the object, it is evident that l ¼ hnouo and is called the Lagrange invariant. So it follows that for the ith surface,

li ¼ yiniui yiniui ¼ hnouo

If the image is located at the kth surface, then yk ¼ 0 and lk ¼ h0nkuk. As shown in the prior chapter, the lateral system magnification is given by

m ¼ h0 ¼ nouo : h nkuk

Using the Lagrange invariant, the image height can be expressed in terms of the axial ray final slope angle and the Lagrange invariant. This is simply

It is evident that the Lagrange invariant can be used to form intermediate images by each surface comprising the system. In other words, the image formed by the first surface of the object becomes the object for the second surface to form an image, and so on until the final image is reached.

Buchdahl recognized that imaging could be achieved by propagating the image surface by a surface utilizing the Lagrange invariant for an astigmatic optical system.3 He then defined the Buchdahl quasi-invariant defined as

L Hnu

where H is the image height of a real ray in contrast to a paraxial ray. In the paraxial limit, L reduces to l. Since L is based on the real ray height at each intermediate image, the aberration at each surface causes the real-ray intermediate image heights to differ from the corresponding paraxial image heights, which is why Buchdahl called L the quasi-invariant. Now, because image height for the ith surface is the same as the object height for the (iþ1)th surface,

Hi0 ¼ Hiþ1

and it is apparent that

L0i ¼ Liþ1:

Consequently, it follows that at the final system image located at the kth surface (image plane),

k

L0k ¼ L1 þ X DLi

i¼1

where D represents the difference between L before and after refraction/ reflection at a surface.

So DLi L0i Li. Using the above definition for L, we obtain

k

X

DLi ¼ H0nkuk Hnouo:

i¼1

Recalling that Hy0 ¼ h0 þ ey and the lateral system magnification definition, it follows that

k

X

DLi ¼ eynkuk:

i¼1

For ex, the Lagrange invariant is zero. The ray aberration can now be defined as follows,

The total ray aberration is the sum of the individual surface contributions. It is important to understand that the surface contributions are related to the final image rather than the intermediate images. Although it is possible to compute the transverse aberration of the intermediate images by using the local marginal ray slope angle niui rather than nkuk, these aberrations are not additive, that is, they may not be added together to get the final image aberration. Computing the transverse aberration at the intermediate images has no practical utility or meaning.

A general skew ray can be specified at the ith surface by its spatial coordinates ðXi; Yi; ZiÞ and direction cosines ðKi; Li; MiÞ. The paraxial ray coordinates ðy; nuÞ can be generalized in the following manner. In the prior chapter, it was shown that the paraxial ray height at a surface is actually the height at the surface tangent plane. In addition, nu is properly interpreted as n tan u. For a meridional ray, the real ray coordinates can be written in a form similar to the paraxial ray coordinates as ðY; UyÞ where

L

Uy M ¼ tan U

Buchdahl referred to the ðY; UyÞ coordinates as canonical coordinates and they can be used for ray tracing as well; however, the prime object is to determine DL for each surface. Although the derivation of DL is tedious, it is straightforward to show that

where DU ¼ Uiþ1 Ui. The change in the Buchdahl quasi-invariant across a surface boundary is given exactly by Eq. (4-19).

The canonical coordinates ðYi; UiÞ are nonlinear functions of the object ray coordinates ðY1; U1Þ. Consequently, the coordinate values needed to solve Eq. (4-19) are unknown. The solution is to perform a series expansion of DL in terms of the canonical coordinates. It can be shown that DL can be expanded as an odd-order polynomial, namely

1 3 5

DL ¼ D L þD L þD L þ . . .

w

where L represents the wth-order of the polynomial expansion of DL. Since

1 1 3 5

L ¼ l, then D L ¼ Dl ¼ 0 and DL ¼ D L þD L þ . . . : This is consistent with the premise that first-order or paraxial optics is aberration free. Now, because the ray aberrations are linearly related to DL, we can write

3 5 7

e ¼ e þ e þ e þ . . . :

which is a statement that the ray aberrations can be expressed as a summation of third, fifth, seventh, and higher orders. Once the expansion is completed, it is observed that the third-order term of DL depends only on the linear part of the approximations of Y and U while the nonlinear parts of these approximations give rise to fifthand higher-order aberrations. Seidel and others realized that the third-order aberrations can be computed using data from only two paraxial rays (marginal and principal).

An orderly iterative process for computing the higher-order aberration terms was achieved by Buchdahl somewhat less than a hundred years after Seidel published his work. As mentioned previously, from Buchdahl’s work and that of others, it became understood that aberration coefficients comprise intrinsic and extrinsic contributions.15 Extrinsic contributions of, say, the ith surface affect the aberration coefficient values of subsequent surfaces while the intrinsic contributions remain local to that surface. Third-order aberration coefficients do not have extrinsic contributions which means these coefficients are decoupled from one another unlike the higher-order aberration coefficients. The nonlinear parts of the approximations of Y and U, and the existence of the extrinsic contributions are reasons the general lens design problem is quite nonlinear and often difficult to optimize.

In actual practice, the lens designer observes that the higher the order of the aberration, the more stable the aberration is with respect to changes in constructional parameters such as curvature and thickness. For example, the values of the third-order aberrations will change much more rapidly, in general, than the fifth-order aberrations if a curvature is changed. It is generally understood by lens designers that if a lens suffers from higher-order aberrations, some significant change to the current optical configuration will be necessary.

With further algebraic effort, DL is transformed into the third-order form of ex and ey which can be written in terms of paraxial entering ray coordinates, ðr; y; HÞ, namely,

The third-order aberration coefficients, s1 through s5, for a given optical system can be calculated using the ray data obtained by tracing the marginal and principal paraxial rays using the following equations. The coefficient form with the presubscript is used to denote the aberration contribution of the ith surface. It is important to understand that these coefficients can be used to compute transverse, longitudinal, and wave aberrations, which are related by scaling factors.

ii ¼ ciyi þ ni 1ui 1 ni 1

is5 ¼ qiðq2i is1 þ is4Þ

The transverse third-order aberration coefficients are determined by summation of the surface contributions and then multiplying by the factor

1

2nkuk

Notice that the Petzval term s4 is also multiplied by the square of the Lagrange invariant, yn 1u 1 yn 1u 1: