ENDNOTES

1After Rudolf Kingslake, Optical System Design, Chapter 5.IV, Academic Press, New York (1983).

2W. Brower, Matrix Methods in Optical Instrument Design, Benjamin, New York (1964).

3H. Kogelnik, “Paraxial ray propagation,” in Applied Optics and Optical Engineering, Vol. 7,

p.156, R. R. Shannon and J. C. Wyant (Eds.), Academic Press, New York (1979).

4Prior to the 1839 Petzval Portrait lens, the term focal length had meaning only for thin lenses, which was taken as the distance from the lens to the image formed when viewing a very distant object. For a long time, the term equivalent focal length was used for a complex lens to mean the focal length of the equivalent thin lens.

5H. Erfle, “Die optische Abbildung durch Kugelflaechen,” Chapter III in S. Czapski und O. Eppenstein Grundzuege der Theorie der Optischen Instrumente nach Abbe, Third Edition,

pp.72–134, H. Erfle and H. Boegehold (Eds.), Barth, Leipzig (1924).

6H. Schroeder, “Notiz betreffend die Gaussischen Hauptpunkte,” Astron. Nachrichten, 111:187–188 (1885).

7R. Barry Johnson, “Correctly making panoramic imagery and the meaning of optical center,” Current Developments in Lens Design and Optical Engineering IX, Pantazis Z. Mouroulis, Warren J. Smith, and R. Barry Johnson (Eds.), Proc. SPIE, 7060:70600F (2008).

8R. Barry Johnson, James B. Hadaway, Tom Burleson, Bob Watts, and Ernest D. Park, “All-reflective four-element zoom telescope: Design and analysis,” International Lens Design Conference, Proc. SPIE, 1354:669–675 (1990).

9R.H.R. Cuvillier, “Le Pan-Cinor et ses applications,” La Tech. Cinemat., 21:73 (1950); also U.S. Patent 2,566,485, filed January 1950.

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axis. Referring to Figure 4.1, this plane is the Y-Z plane and is generally called the tangential or meridional plane.

Assume now that a stigmatic image of the object plane is formed in the image plane where the object has some geometrical shape. If the optical system forms an image having the same geometrical shape as the object to some scaling factor, the image is considered to be undistorted or be an accurate geometric representation of the object. Should the optical system form an image which is not geometrically similar to the object’s shape, then the image is said to suffer distortion. When the system is free of distortion (undistorted), the ratio of image size to the corresponding object size is the magnification m, with the image for a positive lens being inverted and reverted with respect to the object. Let the object be a line extending from the origin of the object plane to the location denoted as point object in Figure 4.1 which

has coordinates expressed as Hx; Hy . The image size can be computed by

H 0x ¼ mHx and H0y ¼ mHy

since the line can be projected onto each axis and propagated independently without loss of generality since a paraxial skew ray is linearly separable into its orthogonal components.

It is evident from the preceding discussion that an ideal image of the object plane requires three conditions to be satisfied, namely, stigmatic image formation, no curvature of field, and no distortion. In contrast, an optical system having stigmatic image formation can still suffer the image defects of distortion and curvature of field.

As explained, an ideal optical system forms a perfect or stigmatic image which essentially means that rays emanating from a point source will be converged by the optical system to a point image, although curvature of field and distortion may be present. At this juncture, image quality will be discussed in strictly geometric terms. In later chapters, the impact of diffraction on image quality will be discussed.

The majority of this book addresses rotationally symmetric optical systems, their aberrations, and configurations. Figure 4.1 shows the generic geometry for such systems, which comprise five principal elements: the object plane, entrance pupil, lenses (including stop), exit pupil, and image plane.4 A ray propagating through this system is specified by its object coordinates ðHx; HyÞ and entrance pupil coordinates ðrx; ryÞ ¼ ~r, or in polar coordinates ðr; yÞ, as illustrated in Figure 4.2. This means that point P in the entrance pupil can be expressed by X ¼ r cosðyÞ and Y ¼ r sinðyÞ where y is zero when ~r lies along the Y-axis.

This ray is incident on the image plane at ðH 0x; H 0yÞ and displaced or aberrant from the ideal image location by ðex; eyÞ. Since the optical system is rotationally symmetric, the (point) object is assumed to always be located on the y-axis in the object plane, that is, H ð0; HyÞ. This means the ideal image is located along the y-axis in the image plane, that is, h0 ¼ mH where m is the magnification. The actual

Hy

P(X,Y )

Y

H

Hx

Z

pupil Entrance

Figure 4.2 Entrance pupil coordinates of a ray.

image plane may be displaced a distance x from the ideal image plane. The ideal image plane is also called the Gaussian or paraxial image plane. The term image plane, as used in this book, means the planar surface where the image is formed which may be displaced from the ideal image plane by the defocus distance x.

A ray exiting the exit pupil, as shown in Figure 4.3, intersects the image plane at ðX 0; Y0Þ which in general does not pass through the ideal image

Actual image

X′

Ideal image

Z

x ex ey

plane Image

Figure 4.3 Image plane coordinates of ray suffering aberrations.

point shown in the figure as a consequence of aberrations. The object point is

0ðr; y; ~ ; xÞ e ðr; y; ~ ; xÞ þ 0

X H x H hx

0ðr; y; ~; xÞ e ðr; y; ~ ; xÞ þ 0

Y H y H hy

Using vector notation and ignoring the defocus parameter for the moment, the ray aberration can be written as

~eð~r; ~Þ ¼~e ð~r; ~ Þ þ~e ð~r; ~ Þ

H s H c H

and~es and ~ec are called the symmetric and asymmetric aberrations as well as the

importance of decomposing the ray aberration in this manner for our study of lens design will become evident. Consider first the symmetric term ~es which means that the ray error will be symmetric about the ideal image location

for an optical system suffering only astigmatic aberration, the pattern formed will be symmetric.

identical image error ðex; eyÞ, that is, they each intercept the image plane at the same location. Consequently, the comatic aberration creates an asymmetry in the spot diagram. Further, it should be recognized that the astigmatic and comatic aberration components are decoupled and can not be used to balance one another. The importance of this knowledge in lens design will be explained in more detail in the following chapters.

Since the optical system is rotationally symmetric, the object can be placed in the meridional plane, or y-axis of the object plane, without the loss of generality and the advantage of simplifying the computation and interpretation of the resulting aberrations. Consequently, since the x-component is zero the object is denoted by H and the ideal image by h0. The actual image coordinates now become

X 0ðr; y; H; xÞ exðr; y; H; xÞ

Y 0ðr; y; H; xÞ eyðr; y; H; xÞ þ h0

It has been found useful to decompose the aberration into two elements with respect to how the aberrations transform under a change in the sign of r. These elements are called symmetric and asymmetric components and are orthogonal to one another. Since the object and image are located in the meridional plane, all rays emanating from the object point having entrance pupil coordinates ~r ¼ ðr; 0oÞ necessarily lie in the meridional plane.6 Consequently, ex ¼ 0. It is common to plot the ray aberration for the meridional fan of rays with r being normalized ( 1 to þ1). The ordinate of the plot is the ray error measured from intercept of the principal ray.

Figure 4.4 provides an example of such a plot. In this case, H 6¼0 to allow illustration of the symmetric and asymmetric components of the ray aberration. As explained above, ~es and ~ec represent these components. In this figure, the comatic and the stigmatic contributions for the total aberration are shown.

Notice that the comatic aberration is symmetric about the r ¼ 0 axis. In other words, any ray pair having entrance pupil coordinates of ðr; 0oÞ and ð r; 0oÞ will have the same ray error, that is, eyðr; 0; HÞ ¼ eyð r; 0; HÞ. In contrast, the astigmatic aberration is asymmetric about the same axis. This means that any ray pair having entrance pupil coordinates of ðr; 0oÞ and ð r; 0oÞ will suffer ray errors of equal and opposite sign, that is, eyðr; 0; HÞ ¼ eyð r; 0; HÞ. Examination of the total aberration curve illustrates that it can be neither symmetric nor asymmetric. In this particular case, both the comatic and astigmatic aberrations comprise thirdand fifth-order terms of opposite signs. The total

r

Figure 4.4 Ray aberration for a meridional fan of rays.

where SPH spherical aberration, AST astigmatism, CMA coma, DIST distortion, and DF defocus. It should be recognized that the comatic component of ex does not contain a distortion term since it is assumed that the object lies in the meridional plane.

Being that the ray intercept error can be described as the linear combination of the astigmatic and comatic contributions, these contributions can be written as a power series in terms of H and r. Several conventions exist for expansion nomenclature; however, most follow that given by Buchdahl. Specifically, an aberration depending on r and H in the combination rn sHs is said to be of the type

. nth order, sth degree coma if (n-s) is even, or

. nth order, (n-s)th degree astigmatism if (n-s) is odd.

For simplicity, the arguments of ex and ey are not explicitly shown unless needed for clarity, defocus is assumed zero, and recalling that the expansions are a function of y for the general skew ray, the expansion of the ray errors are given by

ex ¼ ðs1r3 þ m1r5 þ t1r7 þ . . .Þ sinðyÞ

| {z }

SPHERICAL

þ ðs2r2 þ m3r4 þ t3r6 þ . . .Þ sinð2yÞH

| {z }

LINEAR or CIRCULAR COMA

þ ðm9 sinð2yÞr2 þ ðt9 sinð2yÞ þ t10 sinð4yÞÞr4 þ . . .ÞH3

| {z }

CUBIC COMA

þ ððm5 þ m6 cos ðyÞÞH þ ðt13 þ t14 cos ðyÞÞH þ . . .Þ sinðyÞr

| {z }

CUBIC ASTIGMATISM

þ ððt5 þ t6 cos2ðyÞÞH2 þ . . .Þ sinðyÞr5

| {z }

QUINTIC ASTIGMATISM

þ . . . HIGHER ORDER ABERRATIONS IN TERMS OF r AND H:

and

ey ¼ðs1r3 þ m1r5 þ t1r7 þ . . .ÞcosðyÞ

| {z }

SPHERICAL

þðs2ð2 þ cosð2yÞÞr2 þ ðm2 þ m3 cosð2yÞÞr4 þ ðt2 þ t3 cosð2yÞÞr6 þ . . .ÞH

| {z }

LINEAR or CIRCULAR COMA

þððm7 þ m8 cosð2yÞÞr2 þ ðt7 þ t8 cosð2yÞ þ t10 cosð4yÞÞr4 þ . . .ÞH3

| {z }

CUBIC COMA

þ ððt15 þ t16Þcosð2yÞr2 þ . . .ÞH5

| {z }

QUINTIC COMA

þðð3s3 þ s4ÞH2 þ m10H4 þ t18H6 þ . . .ÞcosðyÞr

| {z }

LINEAR ASTIGMATISM

þððm4 þ m6 cos2ðyÞÞH2 þ ðt11 þ t12 cos2ðyÞÞH4 þ . . .ÞcosðyÞr3

| {z }

CUBIC ASTIGMATISM

þ ððt4 þ t6 cos2ðyÞÞH2 þ . . .ÞcosðyÞr5

| {z }

QUINTIC ASTIGMATISM

þs5H3 þ m12H5 þ t20H7 þ . . .

| {z }

DISTORTION

þ . . . HIGHER ORDER ABERRATIONS IN TERMS OF r AND H:

(4-3)

The five s, twelve m, and twenty t coefficients represent the third-, fifth-, and seventh-order terms, respectively. Even-order terms do not appear as a consequence of the rotational symmetry of the optical system. Further, there are actually five, nine, and 14 independent coefficients for the third-, fifth-, and seventh-order terms, respectively.7

There exist three identities between the m coefficients, and six identities between the t coefficients. These identities take the form of a linear combination of the nth-order coefficients being equal to combinations of products of the lowerorder coefficients. If, for example, all of the third-order coefficients have been corrected to zero, then the following identities for the fifth-order coefficients exist:

m2 23 m3 ¼ 0; m4 m5 m6 ¼ 0; and m7 m8 m9 ¼ 0. Calculation of these coefficients is straightforward, although tedious, using the iterative process developed

by Hans Buchdahl.1 The third-order terms were first popularized by the publication of Seidel and are often referred to as the Seidel aberrations.8 The fifth-order terms were first computed in the early twentieth century.9

In the late 1940s, Buchdahl published his work on how to calculate the coefficients to any arbitrary order. However, recent investigation into the historical work of Joseph Petzval, a Hungarian professor of mathematics at Vienna, has lead to the belief that he had developed in the late 1830s a computational scheme through fifth-order and perhaps to seventh-order for spherical aberration.10 Conrady was well aware of the Petzval sum in addition to Petzval’s greater contributions to optics as evidenced when he wrote:

[Petzval] who investigated the aberrations of oblique pencils about 1840, and apparently arrived at a complete theory not only of the primary, but also of the secondary oblique aberrations; but he never published his methods in any complete form, he lost the priority which undoubtedly would have been his. It is, however, perfectly clear from his occasional brief publications that he had a more accurate knowledge of the profound significance of the Petzval theorem than any of his successors in the investigation of the oblique aberrations for some eighty years after his original discovery.11

Regrettably, the preponderance of his work was lost to posterity. The design and development for today’s optical systems were made possible by theoretical understanding of optical aberrations through the contributions of numerous individuals. Although the subject is still evolving, serious research spans over four centuries.12

As an example, consider a meridional ray intersecting the paraxial image plane, and having entrance pupil coordinates of ðr; 900; H; 0Þ: The ex and ey are given by

ex ¼ ðs1r3 þ m1r5 þ t1r7 þ . . .Þ

| {z }

SPHERICAL

þðs3 þ s4ÞH2 þ m11H4 þ t19H6 þ . . . r

þm5H2 þ t13H4 þ . . . r3

þt5H2 þ . . . r5

| {z }

ASTIGMATISM

þ . . . HIGHER ORDER ABERRATIONS IN TERMS OF r AND H:

and

ey ¼ ðs2r2 þ ðm2 m3Þr4 þ ðt2 t3Þr6 þ . . .ÞH

þðm7 m8Þr2 þ ðt7 t8 þ t10Þr4 þ . . . H3

þðt15 t16Þr2 þ . . . H5

| {z }

COMA

þ s5H3 þ m12H5 þ t20H7 þ . . .

| {z }

DISTORTION

þ . . . HIGHER ORDER ABERRATIONS IN TERMS OF r AND H:

Observe that the sagittal term ex comprises only astigmatic contributions, while the meridional term ey contains only comatic contributions. The ability to isolate specific contributions of the ray error by proper selection of one or more rays will be exploited in the remainder of this chapter.

As previously explained, the actual ray height in the paraxial image plane can be considered to comprise two principal elements: the Gaussian ray height and the ray aberration, as illustrated in the aberration map shown in Figure 4.5 on the next page. The total aberration for a rotationally symmetric optical system comprises two orthogonal components, astigmatic and comatic. The astigmatic aberration is segmented into field independent and dependent components while the comatic aberration is divided into aperture independent and dependent components.

The field-independent astigmatic aberration has two contributions, which are defocus and spherical aberration. The defocus x is linearly dependent on the entrance pupil radius r while the spherical aberration is dependent on the odd orders of third and above of the entrance pupil radius, namely, r3; r5; . . . .

The field-independent astigmatic aberration introduces a uniform aberration or blur over the optical system’s field-of-view.

Field-dependent astigmatic aberrations comprise two contributions which are linear astigmatism and oblique spherical aberration. Both of these aberrations are dependent on even orders of H, namely, H2; H4; . . .. Linear astigmatism is linearly dependent on the entrance pupil radius r while the oblique spherical aberration is dependent on the odd orders of third and above of the entrance pupil radius. It should be noted that the defocus and linear astigmatism comprise the linearly-dependent entrance-pupil-radius components of the astigmatic aberration (r; H0; H2; H4; . . .). In a like manner, spherical and oblique spherical aberrations comprise the higher-order terms in entrance- pupil-radius (r3; r5; . . . ; H0; H2; H4; . . .).

Aperture-independent comatic aberration has two contributions, which are the Gaussian image height and distortion. Although the Gaussian image height is not considered an actual aberration, it is shown in the aberration map in a dashed box since the Gaussian image height is linearly proportional to H and aperture independent. Distortion is also aperture independent, but is dependent on the odd orders of third and above of H, namely, H3; H5; . . . .

Aperture-dependent comatic aberration also has two contributions, which are linear coma and nonlinear coma. Linear coma is linearly dependent on the field angle and on even orders of the entrance pupil radius (r2; r4; . . . ; H). Nonlinear coma has the same entrance pupil radius dependence as does linear coma, but is dependent on the odd orders of third and above of H in the same manner as distortion. Perhaps the most common element of nonlinear coma is referred to as elliptical coma; however, there are many other contributions to the nonlinear comatic aberration.

Actual ray height

Gaussian ray height

Total aberration

Figure 4.5 Ray aberration map showing the astigmatic and comatic elements comprising the total ray aberration.

The aperture-dependent comatic aberration can be viewed as a variation of magnification from one zone to another zone of the entrance pupil. It is also noted that because the astigmatic and comatic contributions are orthogonal, changing the location of the image plane from the paraxial location can impact the resultant astigmatic aberration while having no effect on the comatic contribution of the total aberration. In other words, the defocus can change the astigmatic contribution to the total aberration while having no effect on the comatic contribution. This will be discussed in more detail later in this chapter.

An interesting aspect of the Buchdahl aberration expansion is that the contribution for each coefficient is computed surface by surface and then summed to determine the value of the coefficient at the image plane. For example, s1 is the third-order spherical aberration coefficient. Its value for an optical system comprising n surfaces is computed as

n

X s1 ¼ is1

i¼1

Although there will be no attempt to compute the general set of Buchdahl aberration coefficients in this study, it is important to understand certain aspects of their relationship to the design process. It can be shown that these aberration coefficients have intrinsic and extrinsic contributions. The thirdorder aberration coefficients have only intrinsic contributions, which mean that the value of the aberration coefficients computed for any arbitrary surface are not dependent on the aberration coefficient values for any other surface. For the higher-order aberration coefficients, extrinsic contributions exist in addition to the intrinsic contributions. This means that aberration coefficients for the kth surface are to some extent dependent on the preceding surfaces while not at all dependent on the subsequent surfaces.

Two other characteristics of aberration coefficients are valuable for the lens designer to understand. The first is that lower-order aberration coefficients affect similar high-order aberration coefficients. An alternative way to express this behavior is that higher-order aberration coefficients do not affect the value of lower-order aberration coefficients; that is, adjustment of say t1 does not change the thirdand fifth-order contributions. The second characteristic is that higher-order aberration coefficients move or change their values slowly with changes in constructional parameters (radii, thickness, etc.) compared to the movement of lower-order aberration coefficients. In short, this means that higher-order aberrations, be they astigmatic or comatic, are far more stable than lower-order aberrations.