Note

The rays, depicted as blue lines, are thought of as traveling from left to right. They first appear as if emanating from a black vertical line slightly to the left of the lens. They are moving in straight parallel lines and form a "collimated beam," just as they would from a star located at an infinite (or extremely large) distance to the left. They then travel to the right until meeting our refractor lens, where they are refracted together and converge to a point (focus) on the right-hand side of the screen.

In all the subsequent layouts--whether or not any rays are actually plotted--light is to be thought of as traveling initially from left to right, so that the leftmost surface of any lens is to be thought of as facing the stars. Only in telescopic systems where a mirror is used does the direction of light-travel change. In that case, with every reflection light reverses direction. But in nearly all the cases which we will consider, light is never reflected, but only refracted. Accordingly, almost always the telescope's focus will appear to the right of the objective (or "entrance pupil").

For the purpose of mathematically tracing the paths of light rays through a telescope, designers have evolved systems of equations which depend on coordinate frames notionally attached to lenses and mirrors. The equations themselves will not concern us, but the reader will find it useful to have some grasp of the coordinate frames. A 3-dimensional Cartesian system is used with the z-axis passing through the center of any lens or mirror and running positively to the right and negatively to the left. This line is called the "optical axis" of the telescope. Perpendicular to this axis and running vertically up (positive) and down (negative) is the y-axis. The x-axis runs perpendicularly to the y- and z-axes into and out of the plane of the computer screen (which defines the y/z-plane). X- runs positively into of the computer screen and away from the viewer, negatively out of the screen and toward the viewer. The origin of the coordinate system is not fixed in space, but moves from the center of one lens surface (its so-called "vertex") to the next, following the course of the light rays as the optical designer traces the light paths through the system. That is to say, the ray-tracing analysis does not employ global coordinates frozen in space, but local ones, attached to whichever surface is under analysis at the moment.

It is not crucial for the reader to commit this information to memory. But it is useful to have a grasp of it in order to understand one of the most important diagnostic tools used to evaluate the residual image errors ("aberrations") of lenses and mirrors, namely the "transverse ray fan plots." The following is one such plot:

Clearly the rays toward the margins of this lens come to a shorter focus than those which pass through near the lens's center (the so-called "paraxial" rays). This difference in focus is the defect called "spherical aberration." And because this defect smoothly and symmetrically worsens as we examine rays which enter the lens at greater and greater radial distances for the optical (or "z-") axis (in both positive and negative y- directions), if we were to graph the radial distances along the y-axis at which rays enter the entrance pupil versus how far off the optical axis (again in the y- direction) they fall when they reach the desired focus (defined in Figure 3 as the paraxial focus and marked by the short vertical black line at extreme right of the figure), we would obtain 1/2 of a ray fan plot for this lens. The rays in such a fan, proceeding through the lens in the y/z-plane, are called "tangential" or "meridional" rays, and their graph is called the "tangential" or "meridional ray fan plot." In Figure 2 above, we see the tangential ray fan plot of the left side of the box. Its axes are marked "EY" and "PY." The former stands for transverse "error" along the y-axis; that is to say, how far from the z-axis a ray is when it passes by the focal position. "PY" stands for "pupil" height along the y-axis; that is, how far from the z-axis a ray is when it enters the lens, or "entrance pupil."

But recall that perpendicular to the y/z-plane is the x/z-plane (running into and out of the computer screen). Another fan of rays passing along in this plane can also be analyzed. This is the "sagittal ray fan." Every complete transverse ray fan plot will show both a graph of y- radial distances from the z- optical axis for entering rays, versus y- radial distances from the z-axis for rays in the image surface (where the focus is located), as well as a graph of similar x- radial distances. Such a pair of graphs appears as Figure 2 above. And for the lens in Figure 3 we have the following ray fan plots:

Figure 4: Ray Fan Plots for Figure 3 Lens, Showing Spherical Aberration

Now, because we have been examining a star whose collimated rays enter our lens parallel to the optical axis (i.e. "on-axis"; notice the caption in Figures 2 and 4: "OBJ: 0.000 DEG"), the ray fan in the y/z- tangential plane is identical in shape to the corresponding ray fan in the x/z- sagittal plane, both as regards rays entering the entrance pupil and as regards rays arriving at the image surface. In other words, for every ray with a certain "PY" value, there is a corresponding ray with the same "PX" value. And for every ray arriving in the image surface with a given "EY" value and coming from a certain "PY" position, there is likewise a corresponding ray arriving in the image surface with an identical "EX" value coming from a "PX" position whose value is the same as the previously mentioned "PY" ray. Thus, the tangential and sagittal ray fans are exactly symmetrical so far as the lens is concerned, except that the two fans are rotated by 90 degrees along the optical axis. Accordingly, the tangential and sagittal ray fan plots appear identical to one another, as we see in Figure 4 above.

But for almost all optical systems, if we examine the same two tangential and sagittal ray fan plots for a star which is off-axis--that is, whose rays enter the entrance pupil inclined at an angle--we get two differing graphs, often differing from one another radically. What follows is the same lens as in Figure 3, but with a star 10 degrees off-axis in the meridional plane:

Figure 5: Layout of Lens with Rays from an Off-axis Star

Clearly the rays can no longer form a symmetrical image--at least not symmetrical in the meridional plane. In the sagittal plane, however, the rays progressing through the system still do display symmetry, since their tilt occurs only in the meridional plane. If we could only look down on the lens, instead of viewing it from the side, we would see what happens in the sagittal plane. In fact, ZEMAX will allow us to do that if we view a "3d Layout." Examining that layout looking down from above on to the sagittal plane shows a ray fan which looks indistinguishable from what is seen in Figure 3 above. Thus, in order to conserve space I omit a separate layout looking down on to the sagittal plane. Figure 5, of course, shows the view into the meridional plane.

The following gives the corresponding transverse ray fan plots:

Figure 6: Ray Fan Plots from an Off-axis Star

Notice the caption: "OBJ: 10.00 DEG." Remember that our star is 10 degrees off-axis in the meridional plane. Clearly we can see an asymmetrically shaped meridional plot (on left), corresponding to the asymmetrically shaped meridional ray fan seen in the layout in Figure 5; but a symmetrically shaped sagittal plot (on right) reflecting the symmetrically shaped sagittal ray fan.

In the following chapters we will make extensive use of such ray fan plots to diagnose what image aberrations are present in any given lens. Thus it is important to grasp the significance of the transverse ray fan plots, and to understand how they are formed.

Before proceeding onward, one final set of conventions must be discussed. As I noted above, the z-axis runs positively to the right, and negatively to the left. ZEMAX makes use of this convention in defining precisely how a particular lens is constructed. All lenses are defined by several parameters: their surface curvatures ("radii"), thicknesses, spacings relative to one another, their diameters, glass types, and surface types (i.e. spherical, paraboloidal, hyperboloidal, etc.). The following is a listing of the constructional parameters of the lens shown in Figures 3 and 5:

Surface	Type	Radius	Thickness	Glass	Diameter	Conic
Object	Standard	Infinity	Infinity		0	0
Stop	Standard	400	35	SF57	200	0
2	Standard	-400	229.335		197.6649	0
Image	Standard	Infinity			53.09488	0

Table 1: Prescription of a Lens

The surfaces in the entire system are listed on the left in the order in which light rays encounter them, starting from the object itself, proceeding to the entrance pupil or "stop surface," then through the rest of the optical surfaces, and finally on to the image. In the above lens the surface types are either sections of planes or of spheres. These two lens geometries, along with any type of regular conic surface (i.e. ellipsoids, paraboloids, or hyperboloids) are called "standard" surfaces in ZEMAX. Any other type of surface (for example, a Schmidt plate) has a separate designation.

The radii of curvature for the given surface come next in the listing. Surfaces whose centers lie in the positive z- direction (i.e. surfaces which bulge to the left) are reckoned to have positive radii. And conversely, surfaces whose centers lie in a negative z- direction (i.e. which bulge to the right) are reckoned to have negative radii. Thus, the lens in Figures 3 and 5 has first a convex surface which bulges to the left, and whose center of curvature falls to the right. Therefore, this surface is reckoned positive (+400mm radius of curvature). And secondly the lens has a convex surface which bulges to the right, and whose center of curvature falls to the left. Therefore, this second surface is reckoned negative (-400mm radius of curvature). The lens geometry is thus equiconvex.

Next come the thicknesses (or spacings between surfaces). The object is at infinity, so that the "thickness" from the object to surface 1 is also infinity. The lens itself is 35mm thick. So the distance from surface 1 to surface 2 is 35mm. And the distance, or "thickness," from surface 2 to the image--reckoned to lie at the focus of the rays which pass through the lens at an infinitesimal radial distance away from the optical axis (called the "paraxial" focus)--is 229.335mm. This last number is also called the "back focal length" or "BFL" of the lens.

One glass type is employed, Schott Glass Technology's SF57, a very dense flint glass. The diameter of the beam at each surface is next specified. Initially the beam is 200mm in diameter when it enters the lens as collimated light. By the time it has exited surface 2, it has contracted to 197.6649mm wide. And at the paraxial focus, the total beam width is 53.09488mm. Of course, what you would like in a good lens is a width of 0.000mm at the paraxial focus. But our lens suffers from large amounts of spherical aberration, as we saw in Figure 3.

Finally, all our surfaces (object, lens, and image) are plane or spherical in shape. None of them is an aspheric conic. So their conic constants are all zero. Paraboloids, one may remember from mirror making, have conic constants of -1. Hyperboloids have constants <-1, and prolate ellipsoids have constants between those of spheres and paraboloids, in other words <0 and >-1. Any asphere with a constant >0 is called an oblate ellipsoid. Most practical optics have constants of 0 or less.

Armed with an understanding of the conventions for specifying how an optical system is constructed and of what a transverse ray fan plot is, we can now turn to a description of the various image aberrations which afflict optical systems.